Stochastic Target Games with Controlled Loss
Bruno Bouchard
†
Ludovic Moreau
‡
Marcel Nutz
∗
§
May 14, 2013
Abstract
We study a stochastic game where one player tries to find a strategy such that
the state process reaches a target of controlled-loss-type, no matter which action
is chosen by the other player. We provide, in a general setup, a relaxed geometric
dynamic programming principle for this problem and derive, for the case of a
controlled SDE, the corresponding dynamic programming equation in the sense
of viscosity solutions. As an example, we consider a problem of partial hedging
under Knightian uncertainty.
Keywords Stochastic target; Stochastic game; Geometric dynamic programming principle; Viscosity solution
AMS 2000 Subject Classifications 49N70; 91A23; 91A60; 49L20; 49L25
1
Introduction
We study a stochastic (semi) game of the following form. Given an initial condition
(𝑡, 𝑧) in time and space, we try to find a strategy 𝔲[⋅] such that the controlled state
𝔲[𝜈],𝜈
process 𝑍𝑡,𝑧 (⋅) reaches a certain target at the given time 𝑇 , no matter which control
𝜈 is chosen by the adverse player. The target is specified in terms of expected loss; that
is, we are given a real-valued (“loss”) function ℓ and try to keep the expected loss above
a given threshold 𝑝 ∈ ℝ:
)
[ (
]
𝔲[𝜈],𝜈
ess inf 𝔼 ℓ 𝑍𝑡,𝑧 (𝑇 ) ∣ℱ𝑡 ≥ 𝑝 a.s.
(1.1)
𝜈
Instead of a game, one may also see this as a target problem under Knightian uncertainty; then the adverse player has the role of choosing a worst-case scenario.
Our aim is to describe, for given 𝑡, the set Λ(𝑡) of all pairs (𝑧, 𝑝) such that there
exists a strategy 𝔲 attaining the target. We provide, in a general abstract framework, a
geometric dynamic programming principle (GDP) for this set. To this end, 𝑝 is seen as
an additional state variable and formulated dynamically via a family {𝑀 𝜈 } of auxiliary
martingales with expectation 𝑝, indexed by the adverse controls 𝜈. Heuristically, the
GDP then takes the following form: Λ(𝑡) consists of all (𝑧, 𝑝) such that there exist a
strategy 𝔲 and a family {𝑀 𝜈 } satisfying
(
)
𝔲[𝜈],𝜈
𝑍𝑡,𝑧 (𝜏 ) , 𝑀 𝜈 (𝜏 ) ∈ Λ (𝜏 ) a.s.
∗ We are grateful to Pierre Cardaliaguet for valuable discussions and to the anonymous referees for
careful reading and helpful comments.
† CEREMADE, Université Paris Dauphine and CREST-ENSAE, [email protected].
Research supported by ANR Liquirisk.
‡ Department of Mathematics, ETH Zurich, [email protected].
§ Department of Mathematics, Columbia University, New York, [email protected]. Research supported by NSF Grant DMS-1208985.
1
for all adverse controls 𝜈 and all stopping times 𝜏 ≥ 𝑡. The precise version of the
GDP, stated in Theorem 2.1, incorporates several relaxations that allow us to deal with
various technical problems. In particular, the selection of 𝜀-optimal strategies is solved
by a covering argument which is possible due to a continuity assumption on ℓ and a
relaxation in the variable 𝑝. The martingale 𝑀 𝜈 is constructed from the semimartingale
decomposition of the adverse player’s value process.
Our GDP is tailored such that the dynamic programming equation can be derived in
the viscosity sense. We exemplify this in Theorem 3.4 for the standard setup where the
state process is determined by a stochastic differential equation (SDE) with coefficients
controlled by the two players; however, the general GDP applies also in other situations
such as singular control. The solution of the equation, a partial differential equation
(PDE) in our example, corresponds to the indicator function of (the complement of)
the graph of Λ. In Theorem 3.8, we specialize to a case with a monotonicity condition
that is particularly suitable for pricing problems in mathematical finance. Finally, in
order to illustrate various points made throughout the paper, we consider a concrete
example of pricing an option with partial hedging, according to a loss constraint, in
a model where the drift and volatility coefficients of the underlying are uncertain. In
a worst-case analysis, the uncertainty corresponds to an adverse player choosing the
coefficients; a formula for the corresponding seller’s price is given in Theorem 4.1.
Stochastic target (control) problems with almost-sure constraints, corresponding to
the case where ℓ is an indicator function and 𝜈 is absent, were introduced in [24, 25] as an
extension of the classical superhedging problem [13] in mathematical finance. Stochastic
target problems with controlled loss were first studied in [3] and are inspired by the
quantile hedging problem [12]. The present paper is the first to consider stochastic
target games. The rigorous treatment of zero-sum stochastic differential games was
pioneered by [11], where the mentioned selection problem for 𝜀-optimal strategies was
treated by a discretization and a passage to continuous-time limit in the PDEs. Let
us remark, however, that we have not been able to achieve satisfactory results for our
problem using such techniques. We have been importantly influenced by [7], where
the value functions are defined in terms of essential infima and suprema, and then
shown to be deterministic. The formulation with an essential infimum (rather than an
infimum of suitable expectations) in (1.1) is crucial in our case, mainly because {𝑀 𝜈 }
is constructed by a method of non-Markovian control, which raises the fairly delicate
problem of dealing with one nullset for every adverse control 𝜈.
The remainder of the paper is organized as follows. Section 2 contains the abstract
setup and GDP. In Section 3 we specialize to the case of a controlled SDE and derive
the corresponding PDE, first in the general case and then in the monotone case. The
problem of hedging under uncertainty is discussed in Section 4.
2
Geometric dynamic programming principle
In this section, we obtain our geometric dynamic programming principle (GDP) in an
abstract framework. Some of our assumptions are simply the conditions we need in the
proof of the theorem; we will illustrate later how to actually verify them in a typical
setup.
2.1
Problem statement
We fix a time horizon 𝑇 > 0 and a probability space (Ω, ℱ, ℙ) equipped with a filtration
𝔽 = (ℱ𝑡 )𝑡∈[0,𝑇 ] satisfying the usual conditions of right-continuity and completeness. We
shall consider two sets 𝒰 and 𝒱 of controls; for the sake of concreteness, we assume
that each of these sets consists of stochastic processes on (Ω, ℱ), indexed by [0, 𝑇 ], and
2
with values in some sets 𝑈 and 𝑉 , respectively. Moreover, let 𝔘 be a set of mappings
𝔲 : 𝒱 → 𝒰. Each 𝔲 ∈ 𝔘 is called a strategy and the notation 𝔲[𝜈] will be used for the
control it associates with 𝜈 ∈ 𝒱. In applications, 𝔘 will be chosen to consist of mappings
that are non-anticipating; see Section 3 for an example. Furthermore, we are given a
metric space (𝒵, 𝑑𝒵 ) and, for each (𝑡, 𝑧) ∈ [0, 𝑇 ] × 𝒵 and (𝔲, 𝜈) ∈ 𝔘 × 𝒱, an adapted
𝔲[𝜈],𝜈
𝔲[𝜈],𝜈
càdlàg process 𝑍𝑡,𝑧 (⋅) with values in 𝒵 satisfying 𝑍𝑡,𝑧 (𝑡) = 𝑧. For brevity, we set
𝔲[𝜈],𝜈
𝔲,𝜈
𝑍𝑡,𝑧
:= 𝑍𝑡,𝑧
.
Let ℓ : 𝒵 → ℝ be a Borel-measurable function satisfying
)]
[ ( 𝔲,𝜈
(𝑇 ) < ∞ for all (𝑡, 𝑧, 𝔲, 𝜈) ∈ [0, 𝑇 ] × 𝒵 × 𝔘 × 𝒱.
𝔼 ℓ 𝑍𝑡,𝑧
(2.1)
We interpret ℓ as a loss (or “utility”) function and denote by
[ ( 𝔲,𝜈
)
]
𝐼(𝑡, 𝑧, 𝔲, 𝜈) := 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 , (𝑡, 𝑧, 𝔲, 𝜈) ∈ [0, 𝑇 ] × 𝒵 × 𝔘 × 𝒱
the expected loss given 𝜈 (for the player choosing 𝔲) and by
(𝑡, 𝑧, 𝔲) ∈ [0, 𝑇 ] × 𝒵 × 𝔘
𝐽(𝑡, 𝑧, 𝔲) := ess inf 𝐼(𝑡, 𝑧, 𝔲, 𝜈),
𝜈∈𝒱
the worst-case expected loss. The main object of this paper is the reachability set
{
}
Λ(𝑡) := (𝑧, 𝑝) ∈ 𝒵 × ℝ : there exists 𝔲 ∈ 𝔘 such that 𝐽(𝑡, 𝑧, 𝔲) ≥ 𝑝 ℙ-a.s. .
(2.2)
These are the initial conditions (𝑧, 𝑝) such that starting at time 𝑡, the player choosing 𝔲
can attain an expected loss not worse than 𝑝, regardless of the adverse player’s action 𝜈.
The main aim of this paper is to provide a geometric dynamic programming principle
for Λ(𝑡). For the case without adverse player, a corresponding result was obtained
in [24] for the target problem with almost-sure constraints and in [3] for the problem
with controlled loss.
As mentioned above, the dynamic programming for the problem (2.2) requires the
introduction of a suitable set of martingales starting from 𝑝 ∈ ℝ. This role will be
played by certain families1 {𝑀 𝜈 , 𝜈 ∈ 𝒱} of martingales which should be considered as
additional controls. More precisely, we denote by ℳ𝑡,𝑝 the set of all real-valued (rightcontinuous) martingales 𝑀 satisfying 𝑀 (𝑡) = 𝑝 ℙ-a.s., and we fix a set 𝔐𝑡,𝑝 of families
{𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 ; further assumptions on 𝔐𝑡,𝑝 will be introduced below. Since
these martingales are not present in the original problem (2.2), we can choose 𝔐𝑡,𝑝 to
our convenience; see also Remark 2.2 below.
As usual in optimal control, we shall need to concatenate controls and strategies in
time according to certain events. We use the notation
𝜈 ⊕𝜏 𝜈¯ := 𝜈1[0,𝜏 ] + 𝜈¯1(𝜏,𝑇 ]
for the concatenation of two controls 𝜈, 𝜈¯ ∈ 𝒱 at a stopping time 𝜏 . We also introduce
the set
{
}
{
}
𝜈 =(𝑡,𝜏 ] 𝜈¯ := 𝜔 ∈ Ω : 𝜈𝑠 (𝜔) = 𝜈¯𝑠 (𝜔) for all 𝑠 ∈ (𝑡, 𝜏 (𝜔)] .
Analogous notation is used for elements of 𝒰.
In contrast to the setting of control, strategies can be concatenated only at particular
events and stopping times, as otherwise the resulting strategies would fail to be elements
of 𝔘 (in particular, because they may fail to be non-anticipating, see also Section 3).
Therefore, we need to formalize the events and stopping times which are admissible
1 Of course, there is no mathematical difference between families indexed by 𝒱, like {𝑀 𝜈 , 𝜈 ∈ 𝒱}, and
mappings on 𝒱, like 𝔲. We shall use both notions interchangeably, depending on notational convenience.
3
for this purpose: For each 𝑡 ≤ 𝑇 , we consider a set 𝔉𝑡 whose elements are families
{𝐴𝜈 , 𝜈 ∈ 𝒱} ⊂ ℱ𝑡 of events indexed by 𝒱, as well as a set 𝔗𝑡 whose elements are families
{𝜏 𝜈 , 𝜈 ∈ 𝒱} ⊂ 𝒯𝑡 , where 𝒯𝑡 denotes the set of all stopping times with values in [𝑡, 𝑇 ].
We assume that 𝔗𝑡 contains any deterministic time 𝑠 ∈ [𝑡, 𝑇 ] (seen as a constant family
𝜏 𝜈 ≡ 𝑠, 𝜈 ∈ 𝒱). In practice, the sets 𝔉𝑡 and 𝔗𝑡 will not contain all families of events
and stopping times, respectively; one will impose additional conditions on 𝜈 7→ 𝐴𝜈
and 𝜈 7→ 𝜏 𝜈 that are compatible with the conditions defining 𝔘. Both sets should be
seen as auxiliary objects which make it easier (if not possible) to verify the dynamic
programming conditions below.
2.2
The geometric dynamic programming principle
We can now state the conditions for our main result. The first one concerns the concatenation of controls and strategies.
Assumption (C). The following hold for all 𝑡 ∈ [0, 𝑇 ].
(C1) Fix 𝜈0 , 𝜈1 , 𝜈2 ∈ 𝒱 and 𝐴 ∈ ℱ𝑡 . Then 𝜈 := 𝜈0 ⊕𝑡 (𝜈1 1𝐴 + 𝜈2 1𝐴𝑐 ) ∈ 𝒱.
(C2) Fix (𝔲𝑗 )𝑗≥0 ⊂ 𝔘 and let {𝐴𝜈𝑗 , 𝜈 ∈ 𝒱}𝑗≥1 ⊂ 𝔉𝑡 be such that {𝐴𝜈𝑗 , 𝑗 ≥ 1} forms a
partition of Ω for each 𝜈 ∈ 𝒱. Then 𝔲 ∈ 𝔘 for
∑
𝔲[𝜈] := 𝔲0 [𝜈] ⊕𝑡
𝔲𝑗 [𝜈]1𝐴𝜈𝑗 , 𝜈 ∈ 𝒱.
𝑗≥1
(C3) Let 𝔲 ∈ 𝔘 and 𝜈 ∈ 𝒱. Then 𝔲[𝜈 ⊕𝑡 ⋅] ∈ 𝔘.
(C4) Let {𝐴𝜈 , 𝜈 ∈ 𝒱} ⊂ ℱ𝑡 be a family of events such that 𝐴𝜈1 ∩ {𝜈1 =(0,𝑡] 𝜈2 } =
𝐴𝜈2 ∩ {𝜈1 =(0,𝑡] 𝜈2 } for all 𝜈1 , 𝜈2 ∈ 𝒱. Then {𝐴𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔉𝑡 .
(C5) Let {𝜏 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 . Then {𝜏 𝜈1 ≤ 𝑠} ∩ {𝜈1 =(0,𝑠] 𝜈2 } = {𝜏 𝜈2 ≤ 𝑠} ∩ {𝜈1 =(0,𝑠] 𝜈2 }
ℙ-a.s. for all 𝜈1 , 𝜈2 ∈ 𝒱 and 𝑠 ∈ [𝑡, 𝑇 ].
(C6) Let {𝜏 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 . Then, for all 𝑡 ≤ 𝑠1 ≤ 𝑠2 ≤ 𝑇 , {{𝜏 𝜈 ∈ (𝑠1 , 𝑠2 ]}, 𝜈 ∈ 𝒱} and
{{𝜏 𝜈 ∈
/ (𝑠1 , 𝑠2 ]}, 𝜈 ∈ 𝒱} belong to 𝔉𝑠2 .
The second condition concerns the behavior of the state process.
Assumption (Z). The following hold for all (𝑡, 𝑧, 𝑝) ∈ [0, 𝑇 ] × 𝒵 × ℝ and 𝑠 ∈ [𝑡, 𝑇 ].
𝔲1 ,𝜈
𝔲2 ,𝜈
(Z1) 𝑍𝑡,𝑧
(𝑠)(𝜔) = 𝑍𝑡,𝑧
(𝑠)(𝜔) for ℙ-a.e. 𝜔 ∈ {𝔲1 [𝜈] =(𝑡,𝑠] 𝔲2 [𝜈]}, for all 𝜈 ∈ 𝒱 and
𝔲1 , 𝔲2 ∈ 𝔘.
𝔲,𝜈1
𝔲,𝜈2
(𝑠)(𝜔) for ℙ-a.e. 𝜔 ∈ {𝜈1 =(0,𝑠] 𝜈2 }, for all 𝔲 ∈ 𝔘 and 𝜈1 , 𝜈2 ∈ 𝒱.
(Z2) 𝑍𝑡,𝑧
(𝑠)(𝜔) = 𝑍𝑡,𝑧
(Z3) 𝑀 𝜈1 (𝑠)(𝜔) = 𝑀 𝜈2 (𝑠)(𝜔) for ℙ-a.e. 𝜔 ∈ {𝜈1 =(0,𝑠] 𝜈2 }, for all {𝑀 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝
and 𝜈1 , 𝜈2 ∈ 𝒱.
(Z4) There exists a constant 𝐾(𝑡, 𝑧) ∈ ℝ such that
[
]
𝔲,𝜈
ess sup ess inf 𝔼 ℓ(𝑍𝑡,𝑧
(𝑇 ))∣ℱ𝑡 = 𝐾(𝑡, 𝑧) ℙ-a.s.
𝔲∈𝔘
𝜈∈𝒱
The nontrivial assumption here is, of course, (Z4), stating that (a version of) the
𝔲,𝜈
random variable ess sup𝔲∈𝔘 ess inf 𝜈∈𝒱 𝔼[ℓ(𝑍𝑡,𝑧
(𝑇 ))∣ℱ𝑡 ] is deterministic. For the game
determined by a Brownian SDE as considered in Section 3, this will be true by a result
of [7], which, in turn, goes back to an idea of [21] (see also [16]). An extension to jump
diffusions can be found in [6].
While the above assumptions are fundamental, the following conditions are of technical nature. We shall illustrate later how they can be verified.
4
Assumption (I). Let (𝑡, 𝑧) ∈ [0, 𝑇 ] × 𝒵, 𝔲 ∈ 𝔘 and 𝜈 ∈ 𝒱.
𝔲,𝜈
(I1) There exists an adapted right-continuous process 𝑁𝑡,𝑧
of class (D) such that
[ ( 𝔲,𝜈⊕𝑠 𝜈¯
)
]
𝔲,𝜈
ess inf 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑠 ≥ 𝑁𝑡,𝑧
(𝑠) ℙ-a.s. for all 𝑠 ∈ [𝑡, 𝑇 ].
𝜈
¯∈𝒱
𝔲,𝜈
1
(I2) There exists an adapted right-continuous process 𝐿𝔲,𝜈
𝑡,𝑧 such that 𝐿𝑡,𝑧 (𝑠) ∈ 𝐿 and
[ ( 𝔲⊕𝑠 𝔲¯,𝜈
)
]
ess inf 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑠 ≥ 𝐿𝔲,𝜈
𝑡,𝑧 (𝑠) ℙ-a.s. for all 𝑠 ∈ [𝑡, 𝑇 ].
¯∈𝔘
𝔲
𝔲,𝜈2
1
Moreover, 𝐿𝔲,𝜈
𝑡,𝑧 (𝑠)(𝜔) = 𝐿𝑡,𝑧 (𝑠)(𝜔) for ℙ-a.e. 𝜔 ∈ {𝜈1 =(0,𝑠] 𝜈2 }, for all 𝔲 ∈ 𝒰
and 𝜈1 , 𝜈2 ∈ 𝒱.
Assumption (R). Let (𝑡, 𝑧) ∈ [0, 𝑇 ] × 𝒵.
(R1) Fix 𝑠 ∈ [𝑡, 𝑇 ] and 𝜀 > 0. Then there exist a Borel-measurable partition (𝐵𝑗 )𝑗≥1
of 𝒵 and a sequence (𝑧𝑗 )𝑗≥1 ⊂ 𝒵 such that for all 𝔲 ∈ 𝔘, 𝜈 ∈ 𝒱 and 𝑗 ≥ 1,
[
]
⎫
𝔲,𝜈
𝔼 ℓ(𝑍𝑡,𝑧
(𝑇 ))∣ℱ𝑠 ≥ 𝐼(𝑠, 𝑧𝑗 , 𝔲, 𝜈) − 𝜀,



⎬
[
]
𝔲,𝜈⊕𝑠 𝜈
¯
𝔲,𝜈
ess inf 𝔼 ℓ(𝑍𝑡,𝑧
(𝑇 ))∣ℱ𝑠 ≤ 𝐽(𝑠, 𝑧𝑗 , 𝔲[𝜈 ⊕𝑠 ⋅]) + 𝜀, ℙ-a.s. on {𝑍𝑡,𝑧
(𝑠) ∈ 𝐵𝑗 }.
𝜈
¯∈𝒱



⎭
𝔲,𝜈
𝐾(𝑠, 𝑧𝑗 ) − 𝜀 ≤ 𝐾(𝑠, 𝑍𝑡,𝑧
(𝑠)) ≤ 𝐾(𝑠, 𝑧𝑗 ) + 𝜀
{
(R2) lim
sup
𝛿→0 𝜈∈𝒱,𝜏 ∈𝒯𝑡
ℙ
sup 𝑑𝒵
0≤ℎ≤𝛿
(
𝔲,𝜈
𝑍𝑡,𝑧
(𝜏
+
)
𝔲,𝜈
ℎ), 𝑍𝑡,𝑧
(𝜏 )
}
≥ 𝜀 = 0 for all 𝔲 ∈ 𝔘 and 𝜀 > 0.
Our GDP will be stated in terms of the closure
{
}
(𝑧, 𝑝) ∈ 𝒵 × ℝ : there exist (𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 ) → (𝑡, 𝑧, 𝑝)
Λ̄(𝑡) :=
such that (𝑧𝑛 , 𝑝𝑛 ) ∈ Λ(𝑡𝑛 ) and 𝑡𝑛 ≥ 𝑡 for all 𝑛 ≥ 1
and the uniform interior
{
}
Λ̊𝜄 (𝑡) := (𝑧, 𝑝) ∈ 𝒵 × ℝ : (𝑡′ , 𝑧 ′ , 𝑝′ ) ∈ 𝐵𝜄 (𝑡, 𝑧, 𝑝) implies (𝑧 ′ , 𝑝′ ) ∈ Λ(𝑡′ ) ,
where 𝐵𝜄 (𝑡, 𝑧, 𝑝) ⊂ [0, 𝑇 ] × 𝒵 × ℝ denotes the open ball with center (𝑡, 𝑧, 𝑝) and radius
𝜄 > 0 (with respect to the distance function 𝑑𝒵 (𝑧, 𝑧 ′ ) + ∣𝑝 − 𝑝′ ∣ + ∣𝑡 − 𝑡′ ∣). The relaxation
from Λ to Λ̄ and Λ̊𝜄 essentially allows us to reduce to stopping times with countably
many values in the proof of the GDP and thus to avoid regularity assumptions in the
time variable. We shall also relax the variable 𝑝 in the assertion of (GDP2); this is
inspired by [4] and important for the covering argument in the proof of (GDP2), which,
in turn, is crucial due to the lack of a measurable selection theorem for strategies. Of
course, all our relaxations are tailored such that they will not interfere substantially
with the derivation of the dynamic programming equation; cf. Section 3.
Theorem 2.1. Fix (𝑡, 𝑧, 𝑝) ∈ [0, 𝑇 ] × 𝒵 × ℝ and let Assumptions (C), (Z), (I) and (R)
hold true.
(GDP1) If (𝑧, 𝑝) ∈ Λ(𝑡), then there exist 𝔲 ∈ 𝔘 and {𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 such that
( 𝔲,𝜈
)
𝑍𝑡,𝑧 (𝜏 ) , 𝑀 𝜈 (𝜏 ) ∈ Λ̄ (𝜏 ) ℙ-a.s. for all 𝜈 ∈ 𝒱 and 𝜏 ∈ 𝒯𝑡 .
(GDP2) Let 𝜄 > 0, 𝔲 ∈ 𝔘, {𝑀 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 and {𝜏 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 be such that
( 𝔲,𝜈 𝜈
)
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 𝜈 ) ∈ Λ̊𝜄 (𝜏 𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱,
{
}
{
}
′ −
′
and suppose that 𝑀 𝜈 (𝜏 𝜈 )+ : 𝜈 ∈ 𝒱 and 𝐿𝔲,𝜈
𝑡,𝑧 (𝜏 ) : 𝜈 ∈ 𝒱, 𝜏 ∈ 𝒯𝑡 are uniformly
integrable, where 𝐿𝔲,𝜈
𝑡,𝑧 is as in (I2). Then (𝑧, 𝑝 − 𝜀) ∈ Λ(𝑡) for all 𝜀 > 0.
5
The proof is stated in Sections 2.3 and 2.4 below.
Remark 2.2. We shall see in the proof that the family {𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 in (GDP1)
can actually be chosen to be non-anticipating in the sense of (Z3). However, this will
not be used when (GDP1) is applied to derive the dynamic programming equation.
Whether {𝑀 𝜈 , 𝜈 ∈ 𝒱} is an element of 𝔐𝑡,𝑝 will depend on the definition of the latter
set; in fact, we did not make any assumption about its richness. In many application, it
is possible to take 𝔐𝑡,𝑝 to be the set of all non-anticipating families in ℳ𝑡,𝑝 ; however,
we prefer to leave some freedom for the definition of 𝔐𝑡,𝑝 since this may be useful in
ensuring the uniform integrability required in (GDP2).
We conclude this section with a version of the GDP for the case 𝒵 = ℝ𝑑 , where we
show how to reduce from standard regularity conditions on the state process and the
loss function to the assumptions (R1) and (I).
Corollary 2.3. Let Assumptions (C), (Z) and (R2) hold true. Assume also that ℓ is
continuous and that there exist constants 𝐶 ≥ 0 and 𝑞¯ > 𝑞 ≥ 0 and a locally bounded
function 𝜚 : ℝ𝑑 7→ ℝ+ such that
∣ℓ(𝑧)∣ ≤ 𝐶(1 + ∣𝑧∣𝑞 ),
[ 𝔲¯,¯𝜈
]
ess sup 𝔼 ∣𝑍𝑡,𝑧
(𝑇 )∣𝑞¯∣ℱ𝑡 ≤ 𝜚(𝑧)𝑞¯
(2.3)
ℙ-a.s.
and
(2.4)
(¯
𝔲,¯
𝜈 )∈𝔘×𝒱
[
]
¯,𝜈⊕𝑠 𝜈
¯,𝜈⊕𝑠 𝜈
𝔲⊕𝑠 𝔲
¯
𝔲
¯
𝔲,𝜈
(𝑇 )∣ ∣ℱ𝑠 ≤ 𝐶∣𝑍𝑡,𝑧
ess sup 𝔼 ∣𝑍𝑡,𝑧
(𝑇 ) − 𝑍𝑠,𝑧
(𝑠) − 𝑧 ′ ∣
′
ℙ-a.s.
(2.5)
(¯
𝔲,¯
𝜈 )∈𝔘×𝒱
for all (𝑡, 𝑧) ∈ [0, 𝑇 ] × ℝ𝑑 , (𝑠, 𝑧 ′ ) ∈ [𝑡, 𝑇 ] × ℝ𝑑 and (𝔲, 𝜈) ∈ 𝔘 × 𝒱.
Let (𝑡, 𝑧) ∈ [0, 𝑇 ] × ℝ𝑑 and let {𝜏 𝔲,𝜈 , (𝔲, 𝜈) ∈ 𝔘 × 𝒱} ⊂ 𝒯𝑡 be such that the collection
(𝜏 𝔲,𝜈 ) , (𝔲, 𝜈) ∈ 𝔘 × 𝒱} is uniformly bounded in 𝐿∞ .
𝔲,𝜈
{𝑍𝑡,𝑧
(GDP1’) If (𝑧, 𝑝 + 𝜀) ∈ Λ(𝑡) for some 𝜀 > 0, then there exist 𝔲 ∈ 𝔘 and {𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂
ℳ𝑡,𝑝 such that
( 𝔲,𝜈 𝔲,𝜈
)
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 𝔲,𝜈 ) ∈ Λ̄(𝜏 𝔲,𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱.
(GDP2’) If 𝜄 > 0, 𝔲 ∈ 𝔘 and {𝑀 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 are such that
( 𝔲,𝜈 𝔲,𝜈
)
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 𝔲,𝜈 ) ∈ Λ̊𝜄 (𝜏 𝔲,𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱
and {𝜏 𝔲,𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 , then (𝑧, 𝑝 − 𝜀) ∈ Λ(𝑡) for all 𝜀 > 0.
We remark that Corollary 2.3 is usually applied in a setting where 𝜏 𝔲,𝜈 is the exit
𝔲,𝜈
time of 𝑍𝑡,𝑧
from a given ball, so that the boundedness assumption is not restrictive.
(Some adjustments are needed when the state process admits unbounded jumps; see
also [18].)
2.3
Proof of (GDP1)
We fix 𝑡 ∈ [0, 𝑇 ] and (𝑧, 𝑝) ∈ Λ(𝑡) for the remainder of this proof. By the definition (2.2)
of Λ(𝑡), there exists 𝔲 ∈ 𝔘 such that
𝔼 [𝐺(𝜈)∣ℱ𝑡 ] ≥ 𝑝
ℙ-a.s. for all 𝜈 ∈ 𝒱,
𝔲,𝜈
where 𝐺(𝜈) := ℓ(𝑍𝑡,𝑧
(𝑇 )).
(2.6)
In order to construct the family {𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 of martingales, we consider
𝑆 𝜈 (𝑟) := ess inf 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯)∣ℱ𝑟 ] ,
𝜈
¯∈𝒱
6
𝑡 ≤ 𝑟 ≤ 𝑇.
(2.7)
We shall obtain 𝑀 𝜈 from a Doob-Meyer-type decomposition of 𝑆 𝜈 . This can be seen
as a generalization with respect to [3], where the necessary martingale was trivially
constructed by taking the conditional expectation of the terminal reward.
Step 1: We have 𝑆 𝜈 (𝑟) ∈ 𝐿1 (ℙ) and 𝔼 [𝑆 𝜈 (𝑟)∣ℱ𝑠 ] ≥ 𝑆 𝜈 (𝑠) for all 𝑡 ≤ 𝑠 ≤ 𝑟 ≤ 𝑇 and
𝜈 ∈ 𝒱.
The integrability of 𝑆 𝜈 (𝑟) follows from (2.1) and (I1). To see the submartingale
property, we first show that the family {𝔼[𝐺(𝜈 ⊕𝑟 𝜈¯)∣ℱ𝑟 ], 𝜈¯ ∈ 𝒱} is directed downward.
Indeed, given 𝜈¯1 , 𝜈¯2 ∈ 𝒱, the set
𝐴 := {𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯1 )∣ℱ𝑟 ] ≤ 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯2 )∣ℱ𝑟 ]}
is in ℱ𝑟 ; therefore, 𝜈¯3 := 𝜈 ⊕𝑟 (¯
𝜈1 1𝐴 + 𝜈¯2 1𝐴𝑐 ) is an element of 𝒱 by Assumption (C1).
Hence, (Z2) yields that
𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯3 )∣ℱ𝑟 ]
=
𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯1 )1𝐴 + 𝐺(𝜈 ⊕𝑟 𝜈¯2 )1𝐴𝑐 ∣ℱ𝑟 ]
=
𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯1 )∣ℱ𝑟 ] 1𝐴 + 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯2 )∣ℱ𝑟 ] 1𝐴𝑐
= 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯1 )∣ℱ𝑟 ] ∧ 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯2 )∣ℱ𝑟 ] .
As a result, we can find a sequence (¯
𝜈𝑛 )𝑛≥1 in 𝒱 such that 𝔼[𝐺(𝜈 ⊕𝑟 𝜈¯𝑛 )∣ℱ𝑟 ] decreases
ℙ-a.s. to 𝑆 𝜈 (𝑟); cf. [19, Proposition VI-1-1]. Recalling (2.1) and that 𝑆 𝜈 (𝑟) ∈ 𝐿1 (ℙ),
monotone convergence yields that
[
]
𝔼 [𝑆 𝜈 (𝑟)∣ℱ𝑠 ] = 𝔼 lim 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯𝑛 )∣ℱ𝑟 ] ∣ℱ𝑠
𝑛→∞
=
lim 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯𝑛 )∣ℱ𝑠 ]
𝑛→∞
≥ ess inf 𝔼 [𝐺(𝜈 ⊕𝑟 𝜈¯)∣ℱ𝑠 ]
𝜈
¯∈𝒱
≥ ess inf 𝔼 [𝐺(𝜈 ⊕𝑠 𝜈¯)∣ℱ𝑠 ]
𝜈
¯∈𝒱
= 𝑆 𝜈 (𝑠),
where the last inequality follows from the fact that any control 𝜈 ⊕𝑟 𝜈¯, where 𝜈¯ ∈ 𝒱,
can be written in the form 𝜈 ⊕𝑠 (𝜈 ⊕𝑟 𝜈¯); cf. (C1).
Step 2: There exists a family of càdlàg martingales {𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 such that
𝑆 𝜈 (𝑟) ≥ 𝑀 𝜈 (𝑟) ℙ-a.s. for all 𝑟 ∈ [𝑡, 𝑇 ] and 𝜈 ∈ 𝒱.
Fix 𝜈 ∈ 𝒱. By Step 1, 𝑆 𝜈 (⋅) satisfies the submartingale property. Therefore,
𝑆+ (𝑟)(𝜔) :=
lim
𝑆 𝜈 (𝑢)(𝜔)
for 0 ≤ 𝑟 < 𝑇
𝑢∈(𝑟,𝑇 ]∩ℚ, 𝑢→𝑟
and 𝑆+ (𝑇 ) := 𝑆 𝜈 (𝑇 )
is well defined ℙ-a.s.; moreover, recalling that the filtration 𝔽 satisfies the usual conditions, 𝑆+ is a (right-continuous) submartingale satisfying 𝑆+ (𝑟) ≥ 𝑆 𝜈 (𝑟) ℙ-a.s. for all
𝑟 ∈ [𝑡, 𝑇 ] (c.f. [8, Theorem VI.2]). Let 𝐻 ⊂ [𝑡, 𝑇 ] be the set of points where the function
𝑟 7→ 𝔼[𝑆 𝜈 (𝑟)] is not right-continuous. Since this function is increasing, 𝐻 is at most
countable. (If 𝐻 happens to be the empty set, then 𝑆+ defines a modification of 𝑆 𝜈
and the Doob-Meyer decomposition of 𝑆+ yields the result.) Consider the process
¯
𝑆(𝑟)
:= 𝑆+ (𝑟)1𝐻 𝑐 (𝑟) + 𝑆 𝜈 (𝑟)1𝐻 (𝑟),
𝑟 ∈ [𝑡, 𝑇 ].
The arguments (due to E. Lenglart) in the proof of [8, Theorem 10 of Appendix 1]
¯ )∣ℱ𝜎 ] ≥ 𝑆(𝜎)
¯
show that 𝑆¯ is an optional modification of 𝑆 𝜈 and 𝔼[𝑆(𝜏
for all 𝜎, 𝜏 ∈ 𝒯𝑡
𝔲,𝜈
¯
such that 𝜎 ≤ 𝜏 ; that is, 𝑆 is a strong submartingale. Let 𝑁 = 𝑁𝑡,𝑧
be a right𝜈
continuous process of class (D) as in (I1); then 𝑆 (𝑟) ≥ 𝑁 (𝑟) ℙ-a.s. for all 𝑟 implies
that 𝑆+ (𝑟) ≥ 𝑁 (𝑟) ℙ-a.s. for all 𝑟, and since both 𝑆+ and 𝑁 are right-continuous, this
7
shows that 𝑆+ ≥ 𝑁 up to evanescence. Recalling that 𝐻 is countable, we deduce that
𝑆¯ ≥ 𝑁 up to evanescence, and as 𝑆¯ is bounded from above by the martingale generated
¯ ), we conclude that 𝑆¯ is of class (D).
by 𝑆(𝑇
Now the decomposition result of Mertens [17, Theorem 3] yields that there exist a
¯ and a nondecreasing (not necessarily càdlàg) predictable process
(true) martingale 𝑀
¯
¯
𝐶 with 𝐶(𝑡) = 0 such that
¯ + 𝐶,
¯
𝑆¯ = 𝑀
¯ can be chosen to be càdlàg. We can now define
and in view of the usual conditions, 𝑀
𝜈
¯ −𝑀
¯ (𝑡) + 𝑝 on [𝑡, 𝑇 ] and 𝑀 𝜈 (𝑟) := 𝑝 for 𝑟 ∈ [0, 𝑡), then 𝑀 𝜈 ∈ ℳ𝑡,𝑝 . Noting
𝑀 := 𝑀
¯ (𝑡) = 𝑆(𝑡)
¯ = 𝑆 𝜈 (𝑡) ≥ 𝑝 by (2.6), we see that 𝑀 𝜈 has the required property:
that 𝑀
¯ (𝑟) ≤ 𝑆(𝑟)
¯
𝑀 𝜈 (𝑟) ≤ 𝑀
= 𝑆 𝜈 (𝑟) ℙ-a.s. for all 𝑟 ∈ [𝑡, 𝑇 ].
Step 3: Let 𝜏 ∈ 𝒯𝑡 have countably many values. Then
(
)
𝔲,𝜈
𝐾 𝜏, 𝑍𝑡,𝑧
(𝜏 ) ≥ 𝑀 𝜈 (𝜏 ) ℙ-a.s. for all 𝜈 ∈ 𝒱.
Fix 𝜈 ∈ 𝒱 and 𝜀 > 0, let 𝑀 𝜈 be as in Step 2, and let (𝑡𝑖 )𝑖≥1 be the distinct values
of 𝜏 . By Step 2, we have
[ (
)
]
¯
𝔲,𝜈⊕𝑡 𝜈
𝑀 𝜈 (𝑡𝑖 ) ≤ ess inf 𝔼 ℓ 𝑍𝑡,𝑧 𝑖 (𝑇 ) ∣ℱ𝑡𝑖
ℙ-a.s., 𝑖 ≥ 1.
𝜈
¯∈𝒱
Moreover, (R1) yields that for each 𝑖 ≥ 1, we can find a sequence (𝑧𝑖𝑗 )𝑗≥1 ⊂ 𝒵 and a
Borel partition (𝐵𝑖𝑗 )𝑗≥1 of 𝒵 such that
[ (
)
]
𝔲,𝜈⊕𝑡 𝜈
¯
ess inf 𝔼 ℓ 𝑍𝑡,𝑧 𝑖 (𝑇 ) ∣ℱ𝑡𝑖 (𝜔) ≤ 𝐽(𝑡𝑖 , 𝑧𝑖𝑗 , 𝔲[𝜈 ⊕𝑡𝑖 ⋅])(𝜔) + 𝜀
𝜈
¯∈𝒱
𝔲,𝜈
for ℙ-a.e. 𝜔 ∈ 𝐶𝑖𝑗 := {𝑍𝑡,𝑧
(𝑡𝑖 ) ∈ 𝐵𝑖𝑗 }.
Since (C3) and the definition of 𝐾 in (Z4) yield that 𝐽(𝑡𝑖 , 𝑧𝑖𝑗 , 𝔲[𝜈 ⊕𝑡𝑖 ⋅]) ≤ 𝐾(𝑡𝑖 , 𝑧𝑖𝑗 ), we
conclude by (R1) that
𝔲,𝜈
𝑀 𝜈 (𝑡𝑖 )(𝜔) ≤ 𝐾(𝑡𝑖 , 𝑧𝑖𝑗 ) + 𝜀 ≤ 𝐾(𝑡𝑖 , 𝑍𝑡,𝑧
(𝑡𝑖 )(𝜔)) + 2𝜀 for ℙ-a.e. 𝜔 ∈ 𝐶𝑖𝑗 .
Let 𝐴𝑖 := {𝜏 = 𝑡𝑖 } ∈ ℱ𝜏 . Then (𝐴𝑖 ∩ 𝐶𝑖𝑗 )𝑖,𝑗≥1 forms a partition of Ω and the above
shows that
∑
𝔲,𝜈
𝔲,𝜈
𝑀 𝜈 (𝜏 ) − 2𝜀 ≤
𝐾(𝑡𝑖 , 𝑍𝑡,𝑧
(𝑡𝑖 ))1𝐴𝑖 ∩𝐶𝑖𝑗 = 𝐾(𝜏, 𝑍𝑡,𝑧
(𝜏 )) ℙ-a.s.
𝑖,𝑗≥1
As 𝜀 > 0 was arbitrary, the claim follows.
Step 4: We can now prove (GDP1). Given 𝜏 ∈ 𝒯𝑡 , pick a sequence (𝜏𝑛 )𝑛≥1 ⊂ 𝒯𝑡 such
that each 𝜏𝑛 has countably many values and 𝜏𝑛 ↓ 𝜏 ℙ-a.s. In view of the last statement
of Lemma 2.4 below, Step 3 implies that
( 𝔲,𝜈
)
𝑍𝑡,𝑧 (𝜏𝑛 ), 𝑀 𝜈 (𝜏𝑛 ) − 𝑛−1 ∈ Λ(𝜏𝑛 ) ℙ-a.s. for all 𝑛 ≥ 1.
𝔲,𝜈
However, using that 𝑍𝑡,𝑧
and 𝑀 𝜈 are càdlàg, we have
(
)
(
)
𝔲,𝜈
𝔲,𝜈
𝜏𝑛 , 𝑍𝑡,𝑧
(𝜏𝑛 ), 𝑀 𝜈 (𝜏𝑛 ) − 𝑛−1 → 𝜏, 𝑍𝑡,𝑧
(𝜏 ), 𝑀 𝜈 (𝜏 )
ℙ-a.s. as 𝑛 → ∞,
𝔲,𝜈
so that, by the definition of Λ̄, we deduce that (𝑍𝑡,𝑧
(𝜏 ), 𝑀 𝜈 (𝜏 )) ∈ Λ̄(𝜏 ) ℙ-a.s.
□
Lemma 2.4. Let Assumptions (C2), (C4), (Z1) and (Z4) hold true. For each 𝜀 > 0,
there exists a mapping 𝜇𝜀 : [0, 𝑇 ] × 𝒵 → 𝔘 such that
𝐽 (𝑡, 𝑧, 𝜇𝜀 (𝑡, 𝑧)) ≥ 𝐾(𝑡, 𝑧) − 𝜀
ℙ-a.s. for all (𝑡, 𝑧) ∈ [0, 𝑇 ] × 𝒵.
In particular, if (𝑡, 𝑧, 𝑝) ∈ [0, 𝑇 ] × 𝒵 × ℝ, then 𝐾(𝑡, 𝑧) > 𝑝 implies (𝑧, 𝑝) ∈ Λ(𝑡).
8
Proof. Since 𝐾(𝑡, 𝑧) was defined in (Z4) as the essential supremum of 𝐽(𝑡, 𝑧, 𝔲) over 𝔲,
there exists a sequence (𝔲𝑘 (𝑡, 𝑧))𝑘≥1 ⊂ 𝔘 such that
(
)
sup 𝐽 𝑡, 𝑧, 𝔲𝑘 (𝑡, 𝑧) = 𝐾(𝑡, 𝑧) ℙ-a.s.
(2.8)
𝑘≥1
Set Δ0𝑡,𝑧 := ∅ and define inductively the ℱ𝑡 -measurable sets
∪ 𝑗
{ (
)
} 𝑘−1
Δ𝑘𝑡,𝑧 := 𝐽 𝑡, 𝑧, 𝔲𝑘 (𝑡, 𝑧) ≥ 𝐾(𝑡, 𝑧) − 𝜀 ∖
Δ𝑡,𝑧 ,
𝑘 ≥ 1.
𝑗=0
By (2.8), the family {Δ𝑘𝑡,𝑧 , 𝑘 ≥ 1} forms a partition of Ω. Clearly, each Δ𝑘𝑡,𝑧 (seen as a
constant family) satisfies the requirement of (C4), since it does not depend on 𝜈, and
therefore belongs to 𝔉𝑡 . Hence, after fixing some 𝔲0 ∈ 𝔘, (C2) implies that
∑
𝜇𝜀 (𝑡, 𝑧) := 𝔲0 ⊕𝑡
𝔲𝑘 (𝑡, 𝑧)1Δ𝑘𝑡,𝑧 ∈ 𝔘,
𝑘≥1
while (Z1) ensures that
𝐽 (𝑡, 𝑧, 𝜇𝜀 (𝑡, 𝑧))
=
=
=
[ ( 𝜀
)
]
𝜇 (𝑡,𝑧),𝜈
ess inf 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡
𝜈∈𝒱
⎤
⎡
)
∑ ( 𝔲𝑘 (𝑡,𝑧),𝜈
ess inf 𝔼 ⎣
ℓ 𝑍𝑡,𝑧
(𝑇 ) 1Δ𝑘𝑡,𝑧 ∣ℱ𝑡 ⎦
𝜈∈𝒱
ess inf
𝜈∈𝒱
𝑘≥1
∑
[ ( 𝑘
)
]
𝔲 (𝑡,𝑧),𝜈
𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 1Δ𝑘𝑡,𝑧 ,
𝑘≥1
where the last step used that Δ𝑘𝑡,𝑧 is ℱ𝑡 -measurable. Since
[ ( 𝑘
)
]
𝔲 (𝑡,𝑧),𝜈
𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 ≥ 𝐽(𝑡, 𝑧, 𝔲𝑘 (𝑡, 𝑧))
by the definition of 𝐽, it follows by the definition of {Δ𝑘𝑡,𝑧 , 𝑘 ≥ 1} that
∑ (
)
𝐽 (𝑡, 𝑧, 𝜇𝜀 (𝑡, 𝑧)) ≥
𝐽 𝑡, 𝑧, 𝔲𝑘 (𝑡, 𝑧) 1Δ𝑘𝑡,𝑧 ≥ 𝐾(𝑡, 𝑧) − 𝜀 ℙ-a.s.
𝑘≥1
as required.
Remark 2.5. Let us mention that the GDP could also be formulated using families
of submartingales {𝑆 𝜈 , 𝜈 ∈ 𝒱} rather than martingales. Namely, in (GDP1), these
would be the processes defined by (2.7). However, such a formulation would not be
advantageous for applications as in Section 3, because we would then need an additional
control process to describe the (possibly very irregular) finite variation part of 𝑆 𝜈 .
The fact that the martingales {𝑀 𝜈 , 𝜈 ∈ 𝒱} are actually sufficient to obtain a useful
GDP can be explained heuristically as follows: the relevant situation for the dynamic
programming equation corresponds to the adverse player choosing an (almost) optimal
control 𝜈, and then the value process 𝑆 𝜈 will be (almost) a martingale.
2.4
Proof of (GDP2)
In the sequel, we fix (𝑡, 𝑧, 𝑝) ∈ [0, 𝑇 ] × 𝒵 × ℝ and let 𝜄 > 0, 𝔲 ∈ 𝔘, {𝑀 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 ,
{𝜏 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 and 𝐿𝔲,𝜈
𝑡,𝑧 be as in (GDP2). We shall use the dyadic discretization for
the stopping times 𝜏 𝜈 ; that is, given 𝑛 ≥ 1, we set
∑
𝑡𝑛𝑖+1 1(𝑡𝑛𝑖 ,𝑡𝑛𝑖+1 ] (𝜏 𝜈 ), where 𝑡𝑛𝑖 = 𝑖2−𝑛 𝑇 for 0 ≤ 𝑖 ≤ 2𝑛 .
𝜏𝑛𝜈 =
0≤𝑖≤2𝑛 −1
9
We shall first state the proof under the additional assumption that
𝑀 𝜈 (⋅) = 𝑀 𝜈 (⋅ ∧ 𝜏 𝜈 )
for all 𝜈 ∈ 𝒱.
(2.9)
Step 1: Fix 𝜀 > 0 and 𝑛 ≥ 1. There exists 𝔲𝜀𝑛 ∈ 𝔘 such that
[ ( 𝜀
)
]
(
𝔲 ,𝜈
𝔲,𝜈 𝜈 )
𝔼 ℓ 𝑍𝑡,𝑧𝑛 (𝑇 ) ∣ℱ𝜏𝑛𝜈 ≥ 𝐾 𝜏𝑛𝜈 , 𝑍𝑡,𝑧
(𝜏𝑛 ) − 𝜀 ℙ-a.s. for all 𝜈 ∈ 𝒱.
We fix 𝜀 > 0 and 𝑛 ≥ 1. It follows from (R1) and (C2) that, for each 𝑖 ≤ 2𝑛 , we
can find a Borel partition (𝐵𝑖𝑗 )𝑗≥1 of 𝒵 and a sequence (𝑧𝑖𝑗 )𝑗≥1 ⊂ 𝒵 such that, for all
¯ ∈ 𝔘 and 𝜈 ∈ 𝒱,
𝔲
]
[ ( 𝔲⊕ 𝑛 𝔲¯,𝜈
)
𝑡
¯, 𝜈)(𝜔) − 𝜀 and
(2.10)
𝔼 ℓ 𝑍𝑡,𝑧 𝑖 (𝑇 ) ∣ℱ𝑡𝑛𝑖 (𝜔) ≥ 𝐼(𝑡𝑛𝑖 , 𝑧𝑖𝑗 , 𝔲 ⊕𝑡𝑛𝑖 𝔲
(
)
𝔲,𝜈 𝑛
𝐾 (𝑡𝑛𝑖 , 𝑧𝑖𝑗 ) ≥ 𝐾 𝑡𝑛𝑖 , 𝑍𝑡,𝑧
(𝑡𝑖 )(𝜔) − 𝜀
(2.11)
𝔲,𝜈 𝑛
𝜈
for ℙ-a.e. 𝜔 ∈ 𝐶𝑖𝑗
:= {𝑍𝑡,𝑧
(𝑡𝑖 ) ∈ 𝐵𝑖𝑗 }.
𝜈
∩ {𝜏𝑛𝜈 = 𝑡𝑛𝑖 }, and consider
Let 𝜇𝜀 be as in Lemma 2.4, 𝔲𝜀𝑖𝑗 := 𝜇𝜀 (𝑡𝑛𝑖 , 𝑧𝑖𝑗 ) and 𝐴𝜈𝑖𝑗 := 𝐶𝑖𝑗
the mapping
∑
𝜈 7→ 𝔲𝜀𝑛 [𝜈] := 𝔲[𝜈] ⊕𝜏𝑛𝜈
𝔲𝜀𝑖𝑗 [𝜈]1𝐴𝜈𝑖𝑗 .
𝑗≥1, 𝑖≤2𝑛
𝜈
Note that (Z2) and (C4) imply that {𝐶𝑖𝑗
, 𝜈 ∈ 𝒱}𝑗≥1 ⊂ 𝔉𝑡𝑛𝑖 for each 𝑖 ≤ 2𝑛 . Similarly,
it follows from (C6) and the definition of 𝜏𝑛𝜈 that the families {{𝜏𝑛𝜈 = 𝑡𝑛𝑖 }, 𝜈 ∈ 𝒱} and
{{𝜏𝑛𝜈 = 𝑡𝑛𝑖 }𝑐 , 𝜈 ∈ 𝒱} belong to 𝔉𝑡𝑛𝑖 . Therefore, an induction (over 𝑖) based on (C2)
yields that 𝔲𝜀𝑛 ∈ 𝔘. Using successively (2.10), (Z1), the definition of 𝐽, Lemma 2.4 and
(2.11), we deduce that for ℙ-a.e. 𝜔 ∈ 𝐴𝜈𝑖𝑗 ,
[ ( 𝜀
)
]
𝔲 ,𝜈
𝔼 ℓ 𝑍𝑡,𝑧𝑛 (𝑇 ) ∣ℱ𝜏𝑛𝜈 (𝜔) ≥
≥
(
)
𝐼 𝑡𝑛𝑖 , 𝑧𝑖𝑗 , 𝔲𝜀𝑖𝑗 , 𝜈 (𝜔) − 𝜀
𝐽 (𝑡𝑛𝑖 , 𝑧𝑖𝑗 , 𝜇𝜀 (𝑡𝑛𝑖 , 𝑧𝑖𝑗 )) (𝜔) − 𝜀
𝐾 (𝑡𝑛𝑖 , 𝑧𝑖𝑗 ) − 2𝜀
(
)
𝔲,𝜈 𝑛
≥ 𝐾 𝑡𝑛𝑖 , 𝑍𝑡,𝑧
(𝑡𝑖 )(𝜔) − 3𝜀
(
)
𝔲,𝜈 𝜈
= 𝐾 𝜏𝑛𝜈 (𝜔), 𝑍𝑡,𝑧
(𝜏𝑛 )(𝜔) − 3𝜀.
≥
As 𝜀 > 0 was arbitrary and ∪𝑖,𝑗 𝐴𝜈𝑖𝑗 = Ω ℙ-a.s., this proves the claim.
Step 2: Fix 𝜀 > 0 and 𝑛 ≥ 1. For all 𝜈 ∈ 𝒱, we have
[ ( 𝜀
)
]
𝔲 ,𝜈
𝔼 ℓ 𝑍𝑡,𝑧𝑛 (𝑇 ) ∣ℱ𝜏𝑛𝜈 (𝜔) ≥ 𝑀 𝜈 (𝜏𝑛𝜈 )(𝜔) − 𝜀 for ℙ-a.e. 𝜔 ∈ 𝐸𝑛𝜈 ,
where
{(
)
(
)}
𝔲,𝜈 𝜈
𝔲,𝜈 𝜈
𝐸𝑛𝜈 := 𝜏𝑛𝜈 , 𝑍𝑡,𝑧
(𝜏𝑛 ), 𝑀 𝜈 (𝜏𝑛𝜈 ) ∈ 𝐵𝜄 𝜏 𝜈 , 𝑍𝑡,𝑧
(𝜏 ), 𝑀 𝜈 (𝜏 𝜈 ) .
( 𝔲,𝜈 𝜈
)
Indeed, since )𝑍𝑡,𝑧
(𝜏 ), 𝑀 𝜈 (𝜏 𝜈 ) ∈ Λ̊𝜄 (𝜏 𝜈 ) ℙ-a.s., the definition of Λ̊𝜄 entails that
( 𝔲,𝜈
𝑍𝑡,𝑧 (𝜏𝑛𝜈 ), 𝑀 𝜈 (𝜏𝑛𝜈 ) ∈ Λ(𝜏𝑛𝜈 ) for ℙ-a.e. 𝜔 ∈ 𝐸𝑛𝜈 . This, in turn, means that
(
)
𝔲,𝜈 𝜈
𝐾 𝜏𝑛𝜈 (𝜔), 𝑍𝑡,𝑧
(𝜏𝑛 )(𝜔) ≥ 𝑀 𝜈 (𝜏𝑛𝜈 )(𝜔) for ℙ-a.e. 𝜔 ∈ 𝐸𝑛𝜈 .
Now the claim follows from Step 1. (In all this, we actually have 𝑀 𝜈 (𝜏𝑛𝜈 ) = 𝑀 𝜈 (𝜏 𝜈 )
by (2.9), a fact we do not use here.)
Step 3: Let 𝐿𝜈 := 𝐿𝔲,𝜈
𝑡,𝑧 be the process from (I2). Then
[
]
−
𝐾(𝑡, 𝑧) ≥ 𝑝 − 𝜀 − sup 𝔼 (𝐿𝜈 (𝜏𝑛𝜈 ) − 𝑀 𝜈 (𝜏𝑛𝜈 )) 1(𝐸𝑛𝜈 )𝑐 .
𝜈∈𝒱
10
Indeed, it follows from Step 2 and (I2) that
[ ( 𝜀
)
]
𝔲 ,𝜈
𝔼 ℓ 𝑍𝑡,𝑧𝑛 (𝑇 ) ∣ℱ𝑡
[ [ ( 𝜀
]
)
]
[
]
𝔲 ,𝜈
≥ 𝔼 𝑀 𝜈 (𝜏𝑛𝜈 )1𝐸𝑛𝜈 ∣ℱ𝑡 − 𝜀 + 𝔼 𝔼 ℓ 𝑍𝑡,𝑧𝑛 (𝑇 ) ∣ℱ𝜏𝑛𝜈 1(𝐸𝑛𝜈 )𝑐 ∣ℱ𝑡
[
]
[
]
≥ 𝔼 [𝑀 𝜈 (𝜏𝑛𝜈 )∣ℱ𝑡 ] − 𝔼 𝑀 𝜈 (𝜏𝑛𝜈 )1(𝐸𝑛𝜈 )𝑐 ∣ℱ𝑡 − 𝜀 + 𝔼 𝐿𝜈 (𝜏𝑛𝜈 )1(𝐸𝑛𝜈 )𝑐 ∣ℱ𝑡
[
]
= 𝑝 − 𝜀 + 𝔼 (𝐿𝜈 (𝜏𝑛𝜈 ) − 𝑀 𝜈 (𝜏𝑛𝜈 )) 1(𝐸𝑛𝜈 )𝑐 ∣ℱ𝑡 .
By the definitions of 𝐾 and 𝐽, we deduce that
𝐾(𝑡, 𝑧) ≥ 𝐽(𝑡, 𝑧, 𝔲𝜀𝑛 )
[
]
≥ 𝑝 − 𝜀 + ess inf 𝔼 (𝐿𝜈 (𝜏𝑛𝜈 ) − 𝑀 𝜈 (𝜏𝑛𝜈 )) 1(𝐸𝑛𝜈 )𝑐 ∣ℱ𝑡 .
𝜈∈𝒱
Since 𝐾 is deterministic, we can take expectations on both sides to obtain that
[
]
𝜈
𝐾(𝑡, 𝑧) ≥ 𝑝 − 𝜀 + 𝔼 ess inf 𝔼 [𝑌 ∣ℱ𝑡 ] , where 𝑌 𝜈 := (𝐿𝜈 (𝜏𝑛𝜈 ) − 𝑀 𝜈 (𝜏𝑛𝜈 )) 1(𝐸𝑛𝜈 )𝑐 .
𝜈∈𝒱
The family {𝔼 [𝑌 𝜈 ∣ℱ𝑡 ] , 𝜈 ∈ 𝒱} is directed downward; to see this, use (C1), (Z2), (Z3),
(C5) and the last statement in (I2), and argue as in Step 1 of the proof of (GDP1) in
Section 2.3. It then follows that we can find a sequence (𝜈𝑘 )𝑘≥1 ⊂ 𝒱 such that 𝔼 [𝑌 𝜈𝑘 ∣ℱ𝑡 ]
decreases ℙ-a.s. to ess inf 𝜈∈𝒱 𝔼 [𝑌 𝜈 ∣ℱ𝑡 ], cf. [19, Proposition VI-1-1], so that the claim
follows by monotone convergence.
Step 4: We have
]
[
−
lim sup 𝔼 (𝐿𝜈 (𝜏𝑛𝜈 ) − 𝑀 𝜈 (𝜏𝑛𝜈 )) 1(𝐸𝑛𝜈 )𝑐 = 0 ℙ-a.s.
𝑛→∞ 𝜈∈𝒱
Indeed, since 𝑀 𝜈 (𝜏𝑛𝜈 ) = 𝑀 𝜈 (𝜏 𝜈 ) by (2.9), the uniform integrability assumptions
−
in Theorem 2.1 yield that {(𝐿𝜈 (𝜏𝑛𝜈 ) − 𝑀 𝜈 (𝜏𝑛𝜈 )) : 𝑛 ≥ 1, 𝜈 ∈ 𝒱} is again uniformly
integrable. Therefore, it suffices to prove that sup𝜈∈𝒱 ℙ {(𝐸𝑛𝜈 )𝑐 } → 0. To see this, note
that for 𝑛 large enough, we have ∣𝜏𝑛𝜈 − 𝜏 𝜈 ∣ ≤ 2−𝑛 𝑇 ≤ 𝜄/2 and hence
{ ( 𝔲,𝜈 𝜈
}
𝔲,𝜈 𝜈 )
ℙ {(𝐸𝑛𝜈 )𝑐 } ≤ ℙ 𝑑𝒵 𝑍𝑡,𝑧
(𝜏𝑛 ), 𝑍𝑡,𝑧
(𝜏 ) ≥ 𝜄/2 ,
where we have used that 𝑀 𝜈 (𝜏𝑛𝜈 ) = 𝑀 𝜈 (𝜏 𝜈 ). Using once more that ∣𝜏𝑛𝜈 − 𝜏 𝜈 ∣ ≤ 2−𝑛 𝑇 ,
the claim then follows from (R2).
Step 5: The additional assumption (2.9) entails no loss of generality.
˜ 𝜈 be the stopped martingale 𝑀 𝜈 (⋅ ∧ 𝜏 𝜈 ). Then {𝑀
˜ 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 .
Indeed, let 𝑀
𝜈
𝜈
Moreover, since {𝑀 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 and {𝜏 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 , we see from (Z3) and (C5)
˜ 𝜈 , 𝜈 ∈ 𝒱} again satisfies the property stated in (Z3). Finally, we have that the
that{{𝑀
}
{
}
˜
set 𝑀 𝜈 (𝜏 𝜈 )+ : 𝜈 ∈ 𝒱 is uniformly integrable like 𝑀 𝜈 (𝜏 𝜈 )+ : 𝜈 ∈ 𝒱 , since these
˜ 𝜈 , 𝜈 ∈ 𝒱} satisfies all properties required in (GDP2), and of
sets coincide. Hence, {𝑀
˜ 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 ;
course also (2.9). To be precise, it is not necessarily the case that {𝑀
in fact, we have made no assumption whatsoever about the richness of 𝔐𝑡,𝑝 . However,
the previous properties are all we have used in this proof and hence, we may indeed
˜ 𝜈 for the purpose of proving (GDP2).
replace 𝑀 𝜈 by 𝑀
We can now conclude the proof of (GDP2): in view of Step 4, Step 3 yields that
𝐾(𝑡, 𝑧) ≥ 𝑝 − 𝜀, which by Lemma 2.4 implies the assertion that (𝑧, 𝑝 − 𝜀) ∈ Λ(𝑡).
□
11
2.5
Proof of Corollary 2.3
Step 1: Assume that ℓ is bounded and Lipschitz continuous. Then (I) and (R1) are
satisfied.
Assumption (I) is trivially satisfied; we prove that (2.5) implies Assumption (R1).
Let 𝑡 ≤ 𝑠 ≤ 𝑇 and (𝔲, 𝜈) ∈ 𝔘 × 𝒱. Let 𝑐 be the Lipschitz constant of ℓ. By (2.5), we
have
[ (
]
[
)
(
)
]
𝔲,𝜈
𝔲,𝜈
𝔲,𝜈
𝔲,𝜈
𝔼 ℓ 𝑍𝑡,𝑧 (𝑇 ) − ℓ 𝑍𝑠,𝑧′ (𝑇 ) ∣ℱ𝑠 ≤ 𝑐𝔼 𝑍𝑡,𝑧 (𝑇 ) − 𝑍𝑠,𝑧′ (𝑇 ) ∣ℱ𝑠
𝔲,𝜈
(2.12)
≤ 𝑐𝐶 𝑍𝑡,𝑧
(𝑠) − 𝑧 ′ for all 𝑧, 𝑧 ′ ∈ ℝ𝑑 . Let (𝐵𝑗 )𝑗≥1 be any Borel partition of ℝ𝑑 such that the diameter of
𝐵𝑗 is less than 𝜀/(𝑐𝐶), and let 𝑧𝑗 ∈ 𝐵𝑗 for each 𝑗 ≥ 1. Then
[ (
)
(
)
]
{ 𝔲,𝜈
}
𝔲,𝜈
𝔲,𝜈
(𝑠) ∈ 𝐵𝑗 ,
(𝑇 ) ∣ℱ𝑠 ≤ 𝜀 on 𝐶𝑗𝔲,𝜈 := 𝑍𝑡,𝑧
𝔼 ℓ 𝑍𝑡,𝑧 (𝑇 ) − ℓ 𝑍𝑠,𝑧
𝑗
which implies the first property in (R1). In particular, let 𝜈¯ ∈ 𝒱, then using (C1), we
have
[ (
)
(
)
]
𝔲,𝜈⊕ 𝜈
¯
𝔲,𝜈⊕ 𝜈
¯
𝔲,𝜈⊕𝑠 𝜈
¯
(𝑇
)
∣ℱ
≤ 𝜀 on 𝐶𝑗 𝑠 .
𝔼 ℓ 𝑍𝑡,𝑧 𝑠 (𝑇 ) − ℓ 𝑍𝑠,𝑧
𝑠
𝑗
Since 𝐶𝑗𝔲,𝜈⊕𝑠 𝜈¯ = 𝐶𝑗𝔲,𝜈 by (Z2), we may take the essential infimum over 𝜈¯ ∈ 𝒱 to conclude
that
[ ( 𝔲,𝜈⊕𝑠 𝜈¯
)
]
ess inf 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑠 ≤ 𝐽(𝑠, 𝑧𝑗 , 𝔲[𝜈 ⊕𝑠 ⋅]) + 𝜀 on 𝐶𝑗𝔲,𝜈 ,
𝜈
¯∈𝒱
which is the second property in (R1). Finally, the last property in (R1) is a direct
consequence of (2.12) applied with 𝑡 = 𝑠.
Step 2: We now prove the corollary under the additional assumption that ∣ℓ(𝑧)∣ ≤ 𝐶; we
shall reduce to the Lipschitz case by inf-convolution. Indeed, if we define the functions
ℓ𝑘 by
ℓ𝑘 (𝑧) = inf {ℓ(𝑧 ′ ) + 𝑘∣𝑧 ′ − 𝑧∣}, 𝑘 ≥ 1,
𝑧 ′ ∈ℝ𝑑
then ℓ𝑘 is Lipschitz continuous with Lipschitz constant 𝑘, ∣ℓ𝑘 ∣ ≤ 𝐶, and (ℓ𝑘 )𝑘≥1 converges pointwise to ℓ. Since ℓ is continuous and the sequence (ℓ𝑘 )𝑘≥1 is monotone
increasing, the convergence is uniform on compact sets by Dini’s lemma. That is, for
all 𝑛 ≥ 1,
sup
∣ℓ𝑘 (𝑧) − ℓ(𝑧)∣ ≤ 𝜖𝑛𝑘 ,
(2.13)
𝑧∈ℝ𝑑 , ∣𝑧∣≤𝑛
(𝜖𝑛𝑘 )𝑘≥1
where
is a sequence of numbers such that lim𝑘→∞ 𝜖𝑛𝑘 = 0. Moreover, (2.4)
combined with Chebyshev’s inequality implies that
{ 𝔲,𝜈
}
ess sup ℙ ∣𝑍𝑡,𝑧
(𝑇 )∣ ≥ 𝑛∣ℱ𝑡 ≤ (𝜚(𝑧)/𝑛)𝑞¯.
(2.14)
(𝔲,𝜈)∈𝔘×𝒱
Combining (2.13) and (2.14) and using the fact that ℓ𝑘 − ℓ is bounded by 2𝐶 then leads
to
[ ( 𝔲,𝜈
)
( 𝔲,𝜈
)
]
ess sup 𝔼 ℓ𝑘 𝑍𝑡,𝑧
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 ≤ 𝜖𝑛𝑘 + 2𝐶(𝜚(𝑧)/𝑛)𝑞¯.
(2.15)
(𝔲,𝜈)∈𝔘×𝒱
Let 𝑂 be a bounded subset of ℝ𝑑 , let 𝜂 > 0, and let
[ ( 𝔲,𝜈
)
]
𝐼𝑘 (𝑡, 𝑧, 𝔲, 𝜈) = 𝔼 ℓ𝑘 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 .
(2.16)
Then we can choose an integer 𝑛𝜂𝑂 such that 2𝐶(𝜚(𝑧)/𝑛𝜂𝑂 )𝑞¯ ≤ 𝜂/2 for all 𝑧 ∈ 𝑂 and
𝑛𝜂
𝜂
another integer 𝑘𝑂
such that 𝜖𝑘𝜂𝑂 ≤ 𝜂/2. Under these conditions, (2.15) applied to
𝑂
𝜂
𝑛 = 𝑛𝑂 yields that
𝜂
ess sup 𝐼𝑘𝑂
(𝑡, 𝑧, 𝔲, 𝜈) − 𝐼(𝑡, 𝑧, 𝔲, 𝜈) ≤ 𝜂 for (𝑡, 𝑧) ∈ [0, 𝑇 ] × 𝑂.
(2.17)
(𝔲,𝜈)∈𝔘×𝒱
12
In the sequel, we fix (𝑡, 𝑧, 𝑝) ∈ [0, 𝑇 ] × ℝ𝑑 × ℝ and a bounded set 𝑂 ⊂ ℝ𝑑 containing 𝑧,
𝜂 in terms of ℓ 𝜂 instead of ℓ.
𝜂 , Λ 𝜂 , Λ̊ 𝜂
and define 𝐽𝑘𝑂
𝑘𝑂
𝑘𝑂 ,𝜄 and Λ̄𝑘𝑂
𝑘𝑂
We now prove (GDP1’). To this end, suppose that (𝑧, 𝑝 + 2𝜂) ∈ Λ(𝑡). Then (2.17)
𝜂 (𝑡). In view of Step 1, we may apply (GDP1) with the loss
implies that (𝑧, 𝑝 + 𝜂) ∈ Λ𝑘𝑂
𝜂
function ℓ𝑘𝑂 to obtain 𝔲 ∈ 𝔘 and {𝑀 𝜈 , 𝜈 ∈ 𝒱} ⊂ ℳ𝑡,𝑝 such that
(
)
𝔲,𝜈
𝜂 (𝜏 )
ℙ-a.s. for all 𝜈 ∈ 𝒱 and 𝜏 ∈ 𝒯𝑡 .
𝑍𝑡,𝑧
(𝜏 ), 𝑀 𝜈 (𝜏 ) + 𝜂 ∈ Λ̄𝑘𝑂
Using once more (2.17), we deduce that
( 𝔲,𝜈
)
𝔲,𝜈
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 ) ∈ Λ̄(𝜏 ) ℙ-a.s. for all 𝜈 ∈ 𝒱 and 𝜏 ∈ 𝒯𝑡 such that 𝑍𝑡,𝑧
(𝜏 ) ∈ 𝑂.
𝔲,𝜈
Recalling that {𝑍𝑡,𝑧
(𝜏 𝔲,𝜈 ) , (𝔲, 𝜈) ∈ 𝔘 × 𝒱} is uniformly bounded and enlarging 𝑂 if
necessary, we deduce that (GDP1’) holds for ℓ. (The last two arguments are superfluous
𝜂 already implies Λ̄ 𝜂 (𝜏 ) ⊂ Λ̄(𝜏 ); however, we would like to refer to this proof
as ℓ ≥ ℓ𝑘𝑂
𝑘𝑂
in a similar situation below where there is no monotonicity.)
It remains to prove (GDP2’). To this end, let 𝜄 > 0, 𝔲 ∈ 𝔘, {𝑀 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 and
𝜈
{𝜏 , 𝜈 ∈ 𝒱} ∈ 𝔗𝑡 be such that
( 𝔲,𝜈 𝜈
)
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 𝜈 ) ∈ Λ̊2𝜄 (𝜏 𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱.
For 𝜂 < 𝜄/2, we then have
( 𝔲,𝜈 𝜈
)
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 𝜈 ) + 2𝜂 ∈ Λ̊𝜄 (𝜏 𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱.
(2.18)
𝔲,𝜈
˜ 𝜈 := 𝑀 𝜈 + 𝜂. Since {𝑍𝑡,𝑧
Let 𝑀
(𝜏 𝜈 ) , 𝜈 ∈ 𝒱} is uniformly bounded in 𝐿∞ , we may
𝔲,𝜈 𝜈
assume, by enlarging 𝑂 if necessary, that 𝐵𝜄 (𝑍𝑡,𝑧
(𝜏 )) ⊂ 𝑂 ℙ-a.s. for all 𝜈 ∈ 𝒱. Then,
(2.17) and (2.18) imply that
(
)
𝔲,𝜈 𝜈
˜ 𝜈 (𝜏 𝜈 ) ∈ Λ̊𝑘𝜂 ,𝜄 (𝜏 𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱.
𝑍𝑡,𝑧
(𝜏 ), 𝑀
𝑂
˜ 𝜈 (𝜏 𝜈 ) ≤ 𝐶; in particular, {𝑀
˜ 𝜈 (𝜏 𝜈 )+ , 𝜈 ∈ 𝒱}
Moreover, as ℓ ≤ 𝐶, (2.18) implies that 𝑀
𝔲,𝜈
is uniformly integrable. Furthermore, as ℓ ≥ −𝐶, we can take 𝐿𝑡,𝑧 := −𝐶 for (I2). In
𝜂 then yields that
view of Step 1, (GDP2) applied with the loss function ℓ𝑘𝑂
𝜂 (𝑡)
(𝑧, 𝑝 + 𝜂 − 𝜀) ∈ Λ𝑘𝑂
for all 𝜀 > 0.
(2.19)
˜ 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝+𝜂 , which is not
To be precise, this conclusion would require that {𝑀
necessarily the case under our assumptions. However, since {𝑀 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝 , it is
˜ 𝜈 , 𝜈 ∈ 𝒱} satisfies the property stated in (Z3), so that, as in Step 5 of the
clear that {𝑀
˜ 𝜈 , 𝜈 ∈ 𝒱} ∈ 𝔐𝑡,𝑝+𝜂 .
proof of (GDP2), there is no loss of generality in assuming that {𝑀
We conclude by noting that (2.17) and (2.19) imply that (𝑧, 𝑝 − 𝜀) ∈ Λ(𝑡) for all 𝜀 > 0.
Step 3: We turn to the general case. For 𝑘 ≥ 1, we now define ℓ𝑘 := (ℓ ∧ 𝑘) ∨ (−𝑘),
while 𝐼𝑘 is again defined as in (2.16). We also set
{
}
𝑛𝑘 = max 𝑚 ≥ 0 : 𝐵𝑚 (0) ⊂ {ℓ = ℓ𝑘 } ∧ 𝑘
and note that the continuity of ℓ guarantees that lim𝑘→∞ 𝑛𝑘 = ∞. Given a bounded
set 𝑂 ⊂ ℝ𝑑 and 𝜂 > 0, we claim that
𝜂
ess sup 𝐼𝑘𝑂
(𝑡, 𝑧, 𝔲, 𝜈) − 𝐼(𝑡, 𝑧, 𝔲, 𝜈) ≤ 𝜂 for all (𝑡, 𝑧) ∈ [0, 𝑇 ] × 𝑂
(2.20)
(𝔲,𝜈)∈𝔘×𝒱
13
𝜂
for any large enough integer 𝑘𝑂
. Indeed, let (𝔲, 𝜈) ∈ 𝔘 × 𝒱; then
[
( 𝔲,𝜈
)
]
∣𝐼𝑘 (𝑡, 𝑧, 𝔲, 𝜈) − 𝐼(𝑡, 𝑧, 𝔲, 𝜈)∣ ≤ 𝔼 ∣ℓ − ℓ𝑘 ∣ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡
[
]
( 𝔲,𝜈
)
= 𝔼 ∣ℓ − ℓ𝑘 ∣ 𝑍𝑡,𝑧
(𝑇 ) 1{𝑍 𝔲,𝜈 (𝑇 )∈{ℓ=ℓ
∣ℱ𝑡
/
}
}
𝑘
𝑡,𝑧
[ (
]
)
𝔲,𝜈
≤ 𝔼 ℓ 𝑍𝑡,𝑧 (𝑇 ) 1{∣𝑍 𝔲,𝜈 (𝑇 )∣>𝑛𝑘 } ∣ℱ𝑡
𝑡,𝑧
[(
]
𝔲,𝜈
𝑞 )
≤ 𝐶𝔼 1 + 𝑍𝑡,𝑧 (𝑇 ) 1{∣𝑍 𝔲,𝜈 (𝑇 )∣>𝑛𝑘 } ∣ℱ𝑡
𝑡,𝑧
by (2.3). We may assume that 𝑞 > 0, as otherwise we are in the setting of Step 2. Pick
𝛿 > 0 such that 𝑞(1 + 𝛿) = 𝑞¯. Then Hölder’s inequality and (2.4) yield that
[(
]
)𝑞
𝔲,𝜈
𝔼 𝑍𝑡,𝑧
(𝑇 ) 1{∣𝑍 𝔲,𝜈 (𝑇 )∣>𝑛𝑘 } ∣ℱ𝑡
𝑡,𝑧
1
[(
] 1+𝛿
{ 𝔲,𝜈
} 𝛿
)
𝑞
¯
𝔲,𝜈
ℙ ∣𝑍𝑡,𝑧
(𝑇 )∣ > 𝑛𝑘 ∣ℱ𝑡 1+𝛿
≤ 𝔼 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡
≤
𝑞
¯
𝑞𝛿
¯
𝜌(𝑧) 1+𝛿 (𝜌(𝑧)/𝑛𝑘 ) 1+𝛿 .
Since 𝜌 is locally bounded and lim𝑘→∞ 𝑛𝑘 = ∞, the claim (2.20) follows. We can
then obtain (GDP1’) and (GDP2’) by reducing to the result of Step 2, using the same
arguments as in the proof of Step 2.
□
3
The PDE in the case of a controlled SDE
In this section, we illustrate how our GDP can be used to derive a dynamic programming
equation and how its assumptions can be verified in a typical setup. To this end, we focus
on the case where the state process is determined by a stochastic differential equation
with controlled coefficients; however, other examples could be treated similarly.
3.1
Setup
Let Ω = 𝐶([0, 𝑇 ]; ℝ𝑑 ) be the canonical space of continuous paths equipped with the
Wiener measure ℙ, let 𝔽 = (ℱ𝑡 )𝑡≤𝑇 be the ℙ-augmentation of the filtration generated
by the coordinate-mapping process 𝑊 , and let ℱ = ℱ𝑇 . We define 𝒱, the set of adverse
controls, to be the set of all progressively measurable processes with values in a compact
subset 𝑉 of ℝ𝑑 . Similarly, 𝒰 is the set of all progressively measurable processes with
values in a compact 𝑈 ⊂ ℝ𝑑 . Finally, the set of strategies 𝔘 consists of all mappings
𝔲 : 𝒱 → 𝒰 which are non-anticipating in the sense that
{𝜈1 =(0,𝑠] 𝜈2 } ⊂ {𝔲[𝜈1 ] =(0,𝑠] 𝔲[𝜈2 ]}
for all 𝜈1 , 𝜈2 ∈ 𝒱 and 𝑠 ≤ 𝑇 .
𝔲,𝜈
Given (𝑡, 𝑧) ∈ [0, 𝑇 ] × ℝ𝑑 and (𝔲, 𝜈) ∈ 𝔘 × 𝒱, we let 𝑍𝑡,𝑧
be the unique strong solution
of the controlled SDE
∫ 𝑠
∫ 𝑠
𝑍(𝑠) = 𝑧 +
𝜇(𝑍(𝑟), 𝔲[𝜈]𝑟 , 𝜈𝑟 ) 𝑑𝑟 +
𝜎(𝑍(𝑟), 𝔲[𝜈]𝑟 , 𝜈𝑟 ) 𝑑𝑊𝑟 , 𝑠 ∈ [𝑡, 𝑇 ],
(3.1)
𝑡
𝑡
where the coefficients
𝜇 : ℝ𝑑 × 𝑈 × 𝑉 → ℝ𝑑 ,
𝜎 : ℝ𝑑 × 𝑈 × 𝑉 → ℝ𝑑×𝑑
are assumed to be jointly continuous in all three variables, Lipschitz continuous with
linear growth in the first variable, uniformly in the two last ones, and Lipschitz continuous in the second variable, locally uniformly in the two other ones. Throughout this
section, we assume that ℓ : ℝ𝑑 → ℝ is a continuous function of polynomial growth; i.e.,
14
𝔲,𝜈
(2.3) holds true for some constants 𝐶 and 𝑞. Since 𝑍𝑡,𝑧
(𝑇 ) has moments of all orders,
this implies that the finiteness condition (2.1) is satisfied.
In view of the martingale representation theorem, we can identify the set ℳ𝑡,𝑝 of
martingales
processes 𝛼
∫ with the set 𝒜 of all progressively measurable 𝑑-dimensional
𝛼
such that 𝛼 𝑑𝑊 is a (true) martingale. Indeed, we have ℳ𝑡,𝑝 = {𝑃𝑡,𝑝 , 𝛼 ∈ 𝒜}, where
∫ ⋅
𝛼
𝑃𝑡,𝑝 (⋅) = 𝑝 +
𝛼𝑠 𝑑𝑊𝑠 .
𝑡
We shall denote by 𝔄 the set of all mappings 𝔞[⋅]: 𝒱 7→ 𝒜 such that
{𝜈1 =(0,𝑠] 𝜈2 } ⊂ {𝔞[𝜈1 ] =(0,𝑠] 𝔞[𝜈2 ]}
for all 𝜈1 , 𝜈2 ∈ 𝒱 and 𝑠 ≤ 𝑇 .
𝔞[𝜈]
The set of all families {𝑃𝑡,𝑝 , 𝜈 ∈ 𝒱} with 𝔞 ∈ 𝔄 then forms the set 𝔐𝑡,𝑝 , for any given
(𝑡, 𝑝) ∈ [0, 𝑇 ] × ℝ. Furthermore, 𝔗𝑡 consists of all families {𝜏 𝜈 , 𝜈 ∈ 𝒱} ⊂ 𝒯𝑡 such that,
for some (𝑧, 𝑝) ∈ ℝ𝑑 × ℝ, (𝔲, 𝔞) ∈ 𝔘 × 𝔄 and some Borel set 𝑂 ⊂ [0, 𝑇 ] × ℝ𝑑 × ℝ,
(
)
𝔞[𝜈]
𝔲,𝜈
𝜏 𝜈 is the first exit time of ⋅, 𝑍𝑡,𝑧
, 𝑃𝑡,𝑝 from 𝑂, for all 𝜈 ∈ 𝒱.
(This includes the deterministic times 𝑠 ∈ [𝑡, 𝑇 ] by the choice 𝑂 = [0, 𝑠] × ℝ𝑑 × ℝ.)
Finally, 𝔉𝑡 consists of all families {𝐴𝜈 , 𝜈 ∈ 𝒱} ⊂ ℱ𝑡 such that
𝐴𝜈1 ∩ {𝜈1 =(0,𝑡] 𝜈2 } = 𝐴𝜈2 ∩ {𝜈1 =(0,𝑡] 𝜈2 }
for all 𝜈1 , 𝜈2 ∈ 𝒱.
Proposition 3.1. The conditions of Corollary 2.3 are satisfied in the present setup.
Proof. The above definitions readily yield that Assumptions (C) and (Z1)–(Z3) are
satisfied. Moreover, Assumption (Z4) can be verified exactly as in [7, Proposition 3.3].
Fix any 𝑞¯ > 𝑞 ∨ 2; then (2.4) can be obtained as follows. Let (𝔲, 𝜈) ∈ 𝔘 × 𝒱 and 𝐴 ∈ ℱ𝑡
be arbitrary. Using the Burkholder-Davis-Gundy inequalities, the boundedness of 𝑈
and 𝑉 , and the assumptions on 𝜇 and 𝜎, we obtain that
]
]
[
[
∫ 𝜏
𝔲,𝜈 𝑞¯
𝔲,𝜈 𝑞¯
𝑞¯
sup 𝑍𝑡,𝑧 (𝑠) 1𝐴 𝑑𝑟 ,
𝔼 sup 𝑍𝑡,𝑧 (𝑠) 1𝐴 ≤ 𝑐𝔼 1𝐴 + ∣𝑧∣ 1𝐴 +
𝑡≤𝑠≤𝜏
𝑡
𝑡≤𝑠≤𝑟
𝔲,𝜈
where 𝑐 is a universal constant and 𝜏 is any stopping time such that 𝑍𝑡,𝑧
(⋅ ∧ 𝜏 ) is
bounded. Applying Gronwall’s inequality and letting 𝜏 → 𝑇 , we deduce that
[
]
[
𝑞¯ ]
𝔲,𝜈 𝑞¯
[
]
𝑞¯
𝔲,𝜈
𝔼 𝑍𝑡,𝑧 (𝑇 ) 1𝐴 ≤ 𝔼 sup 𝑍𝑡,𝑧 (𝑢) 1𝐴 ≤ 𝑐𝔼 (1 + ∣𝑧∣ )1𝐴 .
𝑡≤𝑢≤𝑇
Since 𝐴 ∈ ℱ𝑡 was arbitrary, this implies (2.4). To verify the condition (2.5), we note
that the flow property yields
]
]
[
[
𝔲⊕𝑠 𝔲¯,𝜈⊕𝑠 𝜈¯
𝔲¯,𝜈⊕𝑠 𝜈¯
¯,𝜈⊕𝑠 𝜈
¯,𝜈⊕𝑠 𝜈
𝔲
¯
𝔲
¯
𝔼 𝑍𝑡,𝑧
(𝑇 ) − 𝑍𝑠,𝑧
(𝑇 ) 1𝐴 = 𝔼 𝑍𝑠,𝑍
(𝑇
)
−
𝑍
(𝑇
)
1𝐴
𝔲,𝜈
′
′
𝑠,𝑧
(𝑠)
𝑡,𝑧
and estimate the right-hand side with the above arguments. Finally, the same arguments
can be used to verify (R2).
Remark 3.2. We emphasize that our definition of a strategy 𝔲 ∈ 𝔘 does not include
regularity assumptions on the mapping 𝜈 7→ 𝔲[𝜈]. This is in contrast to [2], where a
continuity condition is imposed, enabling the authors to deal with the selection problem
for strategies in the context of a stochastic differential game and use the traditional
formulation of the value functions in terms of infima (not essential infima) and suprema.
Let us mention, however, that such regularity assumptions may preclude existence of
optimal strategies in concrete examples (see also Remark 4.3).
15
3.2
PDE for the reachability set Λ
In this section, we show how the PDE for the reachability set Λ from (2.2) can be
deduced from the geometric dynamic programming principle of Corollary 2.3. This
equation is stated in terms of the indicator function of the complement of the graph
of Λ,
{
0 if (𝑧, 𝑝) ∈ Λ(𝑡)
𝜒(𝑡, 𝑧, 𝑝) := 1 − 1Λ(𝑡) (𝑧, 𝑝) =
1 otherwise,
and its lower semicontinuous envelope
𝜒∗ (𝑡, 𝑧, 𝑝) :=
lim inf
(𝑡′ ,𝑧 ′ ,𝑝′ )→(𝑡,𝑧,𝑝)
𝜒(𝑡′ , 𝑧 ′ , 𝑝′ ).
Corresponding results for the case without adverse player have been obtain in [3, 25];
we extend their arguments to account for the presence of 𝜈 and the fact that we only
have a relaxed GDP. We begin by rephrasing Corollary 2.3 in terms of 𝜒.
Lemma 3.3. Fix (𝑡, 𝑧, 𝑝) ∈ [0, 𝑇 ] × ℝ𝑑 × ℝ and let 𝑂 ⊂ [0, 𝑇 ] × ℝ𝑑 × ℝ be a bounded
open set containing (𝑡, 𝑧, 𝑝).
(GDP1𝜒 ) Assume that 𝜒(𝑡, 𝑧, 𝑝 + 𝜀) = 0 for some 𝜀 > 0. Then there exist 𝔲 ∈ 𝔘 and
{𝛼𝜈 , 𝜈 ∈ 𝒱} ⊂ 𝒜 such that
(
)
𝔲,𝜈 𝜈
𝛼𝜈
𝜒∗ 𝜏 𝜈 , 𝑍𝑡,𝑧
(𝜏 ), 𝑃𝑡,𝑝
(𝜏 𝜈 ) = 0 ℙ-a.s. for all 𝜈 ∈ 𝒱,
( 𝔲,𝜈 𝛼𝜈 )
where 𝜏 𝜈 denotes the first exit time of ⋅, 𝑍𝑡,𝑧
, 𝑃𝑡,𝑝 from 𝑂.
(GDP2𝜒 ) Let 𝜑 be a continuous function such that 𝜑 ≥ 𝜒 and let (𝔲, 𝔞) ∈ 𝔘 × 𝔄 and
𝜂 > 0 be such that
)
(
𝔞[𝜈]
𝔲,𝜈 𝜈
(3.2)
(𝜏 ), 𝑃𝑡,𝑝 (𝜏 𝜈 ) ≤ 1 − 𝜂 ℙ-a.s. for all 𝜈 ∈ 𝒱,
𝜑 𝜏 𝜈 , 𝑍𝑡,𝑧
( 𝔲,𝜈 𝔞[𝜈] )
where 𝜏 𝜈 denotes the first exit time of ⋅, 𝑍𝑡,𝑧
, 𝑃𝑡,𝑝 from 𝑂. Then 𝜒(𝑡, 𝑧, 𝑝 − 𝜀) = 0
for all 𝜀 > 0.
Proof. After observing that (𝑧, 𝑝 + 𝜀) ∈ Λ(𝑡) if and only if 𝜒(𝑡, 𝑧, 𝑝 + 𝜀) = 0 and that
(𝑧, 𝑝) ∈ Λ̄(𝑡) implies 𝜒∗ (𝑡, 𝑧, 𝑝) = 0, (GDP1𝜒 ) follows from Corollary 2.3, whose conditions are satisfied by Proposition 3.1. We now prove (GDP2𝜒 ). Since 𝜑 is continuous
and ∂𝑂 is compact, we can find 𝜄 > 0 such that
𝜑<1
on a 𝜄-neighborhood of ∂𝑂 ∩ {𝜑 ≤ 1 − 𝜂}.
As 𝜒 ≤ 𝜑, it follows that (3.2) implies
( 𝔲,𝜈 𝜈
)
𝑍𝑡,𝑧 (𝜏 ), 𝑀 𝜈 (𝜏 𝜈 ) ∈ Λ̊𝜄 (𝜏 𝜈 ) ℙ-a.s. for all 𝜈 ∈ 𝒱.
Now Corollary 2.3 yields that (𝑧, 𝑝 − 𝜀) ∈ Λ(𝑡); i.e., 𝜒(𝑡, 𝑧, 𝑝 − 𝜀) = 0.
Given a suitably differentiable function 𝜑 = 𝜑(𝑡, 𝑧, 𝑝) on [0, 𝑇 ] × ℝ𝑑+1 , we shall
denote by ∂𝑡 𝜑 its derivative with respect to 𝑡 and by 𝐷𝜑 and 𝐷2 𝜑 the Jacobian and
the Hessian matrix with respect to (𝑧, 𝑝), respectively. Given 𝑢 ∈ 𝑈 , 𝑎 ∈ ℝ𝑑 and 𝑣 ∈ 𝑉 ,
we can then define the Dynkin operator
]
1 [
⊤
⊤
2
ℒ𝑢,𝑎,𝑣
𝜑
:=
∂
𝜑
+
𝜇
(⋅,
𝑢,
𝑣)
𝐷𝜑
+
Tr
𝜎
𝜎
(⋅,
𝑢,
𝑎,
𝑣)𝐷
𝜑
𝑡
(𝑍,𝑃 )
(𝑍,𝑃 ) (𝑍,𝑃 )
(𝑍,𝑃 )
2
with coefficients
(
𝜇(𝑍,𝑃 ) :=
𝜇
0
)
(
,
𝜎(𝑍,𝑃 ) (⋅, 𝑎, ⋅) :=
16
𝜎
𝑎
)
.
To introduce the associated relaxed Hamiltonians, we first define the relaxed kernel
}
{
⊤
≤
𝜀
, 𝜀≥0
(𝑧,
𝑢,
𝑎,
𝑣)𝑞
𝒩𝜀 (𝑧, 𝑞, 𝑣) = (𝑢, 𝑎) ∈ 𝑈 × ℝ𝑑 : 𝜎(𝑍,𝑃
)
for 𝑧 ∈ ℝ𝑑 , 𝑞 ∈ ℝ𝑑+1 and 𝑣 ∈ 𝑉 , as well as the set 𝑁𝐿𝑖𝑝 (𝑧, 𝑞) of all continuous functions
(ˆ
𝑢, 𝑎
ˆ) : ℝ𝑑 × ℝ𝑑+1 × 𝑉 → 𝑈 × ℝ𝑑 ,
(𝑧 ′ , 𝑞 ′ , 𝑣 ′ ) 7→ (ˆ
𝑢, 𝑎
ˆ)(𝑧 ′ , 𝑞 ′ , 𝑣 ′ )
that are locally Lipschitz continuous in (𝑧 ′ , 𝑞 ′ ), uniformly in 𝑣 ′ , and satisfy
(ˆ
𝑢, 𝑎
ˆ) ∈ 𝒩0
on 𝐵 × 𝑉,
for some neighborhood 𝐵 of (𝑧, 𝑞).
The local Lipschitz continuity will be used to ensure the local wellposedness of the SDE
for a Markovian strategy defined via (ˆ
𝑢, 𝑎
ˆ). Setting
{
]}
1 [
⊤
𝐹 (Θ, 𝑢, 𝑎, 𝑣) := −𝜇(𝑍,𝑃 ) (𝑧, 𝑢, 𝑣)⊤ 𝑞 − Tr 𝜎(𝑍,𝑃 ) 𝜎(𝑍,𝑃
(𝑧,
𝑢,
𝑎,
𝑣)𝐴
)
2
for Θ = (𝑧, 𝑞, 𝐴) ∈ ℝ𝑑 × ℝ𝑑+1 × 𝕊𝑑+1 and (𝑢, 𝑎, 𝑣) ∈ 𝑈 × ℝ𝑑 × 𝑉 , we can then define the
relaxed Hamiltonians
𝐻 ∗ (Θ)
:=
𝐻∗ (Θ)
:=
inf lim sup
sup
𝑣∈𝑉 𝜀↘0,Θ′ →Θ (𝑢,𝑎)∈𝒩𝜀 (Θ′ ,𝑣)
sup
𝐹 (Θ′ , 𝑢, 𝑎, 𝑣),
inf 𝐹 (Θ, 𝑢
ˆ(Θ, 𝑣), 𝑎
ˆ(Θ, 𝑣), 𝑣).
(ˆ
𝑢,ˆ
𝑎)∈𝑁𝐿𝑖𝑝 (Θ) 𝑣∈𝑉
(3.3)
(3.4)
(In (3.4), it is not necessary to take the relaxation Θ′ → Θ because inf 𝑣∈𝑉 𝐹 is already
lower semicontinuous.) The question whether 𝐻 ∗ = 𝐻∗ is postponed to the monotone
setting of the next section; see Remark 3.9.
We are now in the position to derive the PDE for 𝜒; in the following, we write
∗
𝐻 𝜑(𝑡, 𝑧, 𝑝) for 𝐻 ∗ (𝑧, 𝐷𝜑(𝑡, 𝑧, 𝑝), 𝐷2 𝜑(𝑡, 𝑧, 𝑝)), and similarly for 𝐻∗ .
Theorem 3.4. The function 𝜒∗ is a viscosity supersolution on [0, 𝑇 ) × ℝ𝑑+1 of
(−∂𝑡 + 𝐻 ∗ )𝜑 ≥ 0.
The function 𝜒∗ is a viscosity subsolution on [0, 𝑇 ) × ℝ𝑑+1 of
(−∂𝑡 + 𝐻∗ )𝜑 ≤ 0.
Proof. Step 1: 𝜒∗ is a viscosity supersolution.
Let (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ∈ [0, 𝑇 ) × ℝ𝑑 × ℝ and let 𝜑 be a smooth function such that
(strict)
min
[0,𝑇 )×ℝ𝑑 ×ℝ
(𝜒∗ − 𝜑) = (𝜒∗ − 𝜑) (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0.
(3.5)
We suppose that
(−∂𝑡 + 𝐻 ∗ )𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ≤ −2𝜂 < 0
(3.6)
for some 𝜂 > 0 and work towards a contradiction. Using the continuity of 𝜇 and 𝜎 and
the definition of the upper-semicontinuous operator 𝐻 ∗ , we can find 𝑣𝑜 ∈ 𝑉 and 𝜀 > 0
such that
𝑜
−ℒ𝑢,𝑎,𝑣
(𝑍,𝑃 ) 𝜑(𝑡, 𝑧, 𝑝) ≤ −𝜂
(3.7)
for all (𝑢, 𝑎) ∈ 𝒩𝜀 (𝑧, 𝐷𝜑(𝑡, 𝑧, 𝑝), 𝑣𝑜 ) and (𝑡, 𝑧, 𝑝) ∈ 𝐵𝜀 ,
where 𝐵𝜀 := 𝐵𝜀 (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) denotes the open ball of radius 𝜀 around (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ). Let
∂𝐵𝜀 := {𝑡𝑜 + 𝜀} × 𝐵𝜀 (𝑧𝑜 , 𝑝𝑜 ) ∪ [𝑡𝑜 , 𝑡𝑜 + 𝜀) × ∂𝐵𝜀 (𝑧𝑜 , 𝑝𝑜 )
17
denote the parabolic boundary of 𝐵𝜀 and set
𝜁 := min(𝜒∗ − 𝜑).
∂𝐵𝜀
In view of (3.5), we have 𝜁 > 0.
Next, we claim that there exists a sequence (𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 , 𝜀𝑛 )𝑛≥1 ⊂ 𝐵𝜀 × (0, 1) such
that
(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 , 𝜀𝑛 ) → (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 , 0)
and 𝜒(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 + 𝜀𝑛 ) = 0
for all 𝑛 ≥ 1.
(3.8)
In view of 𝜒 ∈ {0, 1}, it suffices to show that
𝜒∗ (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0.
(3.9)
Suppose that 𝜒∗ (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) > 0, then the lower semicontinuity of 𝜒∗ yields that 𝜒∗ > 0
and therefore 𝜒 = 1 on a neighborhood of (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ), which implies that 𝜑 has a strict
local maximum in (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) and thus
∂𝑡 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ≤ 0,
𝐷𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0,
𝐷2 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ≤ 0.
This clearly contradicts (3.7), and so the claim follows.
For any 𝑛 ≥ 1, the equality in (3.8) and (GDP1𝜒 ) of Lemma 3.3 yield 𝔲𝑛 ∈ 𝔘 and
𝑛,𝜈
{𝛼 , 𝜈 ∈ 𝒱} ⊂ 𝒜 such that
𝜒∗ (𝑡 ∧ 𝜏𝑛 , 𝑍 𝑛 (𝑡 ∧ 𝜏𝑛 ), 𝑃 𝑛 (𝑡 ∧ 𝜏𝑛 )) = 0,
where
𝑡 ≥ 𝑡𝑛 ,
(3.10)
( 𝑛
)
𝑛,𝑣𝑜
(𝑍 𝑛 (𝑠), 𝑃 𝑛 (𝑠)) := 𝑍𝑡𝔲𝑛 ,𝑧,𝑣𝑛𝑜 (𝑠), 𝑃𝑡𝛼𝑛 ,𝑝𝑛 (𝑠)
and
𝜏𝑛 := inf {𝑠 ≥ 𝑡𝑛 : (𝑠, 𝑍 𝑛 (𝑠), 𝑃 𝑛 (𝑠)) ∈
/ 𝐵𝜀 } .
(In the above, 𝑣𝑜 ∈ 𝑉 is viewed as a constant element of 𝒱.) By (3.10), (3.5) and the
definitions of 𝜁 and 𝜏𝑛 ,
−𝜑(⋅, 𝑍 𝑛 , 𝑃 𝑛 )(𝑡 ∧ 𝜏𝑛 ) = (𝜒∗ − 𝜑)(⋅, 𝑍 𝑛 , 𝑃 𝑛 )(𝑡 ∧ 𝜏𝑛 ) ≥ 𝜁1{𝑡≥𝜏𝑛 } ≥ 0.
Applying Itô’s formula to −𝜑(⋅, 𝑍 𝑛 , 𝑃 𝑛 ), we deduce that
∫ 𝑡∧𝜏𝑛
∫ 𝑡∧𝜏𝑛
𝛿𝑛 (𝑟) 𝑑𝑟 +
Σ𝑛 (𝑟) 𝑑𝑊𝑟 ≥ −𝜁1{𝑡<𝜏𝑛 } ,
𝑆𝑛 (𝑡) := 𝑆𝑛 (0) +
𝑡𝑛
(3.11)
𝑡𝑛
where
𝑆𝑛 (0)
𝛿𝑛 (𝑟)
Σ𝑛 (𝑟)
:=
−𝜁 − 𝜑(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 ),
𝔲𝑛 [𝑣 ],𝛼𝑟𝑛,𝑣𝑜 ,𝑣𝑜
𝑜
𝑟
:= −ℒ(𝑍,𝑃
)
𝜑 (𝑟, 𝑍 𝑛 (𝑟), 𝑃 𝑛 (𝑟)) ,
⊤
:= −𝐷𝜑 (𝑟, 𝑍 𝑛 (𝑟), 𝑃 𝑛 (𝑟)) 𝜎(𝑍,𝑃 ) (𝑍 𝑛 (𝑟), 𝔲𝑛𝑟 [𝑣𝑜 ], 𝛼𝑟𝑛,𝑣𝑜 , 𝑣𝑜 ) .
Define the set
𝐴𝑛 := [[𝑡𝑛 , 𝜏𝑛 ]] ∩ {𝛿𝑛 > −𝜂};
then (3.7) and the definition of 𝒩𝜀 imply that
∣Σ𝑛 ∣ > 𝜀 on 𝐴𝑛 .
Lemma 3.5. After diminishing 𝜀 > 0 if necessary, the stochastic exponential
( ∫ ⋅∧𝜏𝑛
)
𝛿𝑛 (𝑟)
𝐸𝑛 (⋅) = ℰ −
Σ𝑛 (𝑟)1𝐴𝑛 (𝑟) 𝑑𝑊𝑟
∣Σ𝑛 (𝑟)∣2
𝑡𝑛
is well-defined and a true martingale for all 𝑛 ≥ 1.
18
(3.12)
This lemma is proved below; it fills a gap in the previous literature. Admitting its
result for the moment, integration by parts yields
∫ 𝑡∧𝜏𝑛
(𝐸𝑛 𝑆𝑛 )(𝑡 ∧ 𝜏𝑛 ) = 𝑆𝑛 (0) +
𝐸𝑛 𝛿𝑛 1𝐴𝑐𝑛 𝑑𝑟
𝑡𝑛
∫
𝑡∧𝜏𝑛
(
𝐸𝑛
+
𝑡𝑛
𝛿𝑛
Σ𝑛 − 𝑆𝑛
Σ𝑛 1𝐴𝑛
∣Σ𝑛 ∣2
)
𝑑𝑊.
As 𝐸𝑛 ≥ 0, it then follows from the definition of 𝐴𝑛 that 𝐸𝑛 𝛿𝑛 1𝐴𝑐𝑛 ≤ 0 and so 𝐸𝑛 𝑆𝑛
is a local supermartingale; in fact, it is a true supermartingale since it is bounded from
below by the martingale −𝜁𝐸𝑛 . In view of (3.11), we deduce that
[
]
−𝜁 − 𝜑(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 ) = (𝐸𝑛 𝑆𝑛 )(𝑡𝑛 ) ≥ 𝔼 [(𝐸𝑛 𝑆𝑛 )(𝜏𝑛 )] ≥ −𝜁𝔼 1{𝜏𝑛 <𝜏𝑛 } 𝐸𝑛 (𝜏𝑛 ) = 0,
which yields a contradiction due to 𝜁 > 0 and the fact that, by (3.9),
𝜑(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 ) → 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 𝜒∗ (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0.
Step 2: 𝜒∗ is a viscosity subsolution.
Let (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ∈ [0, 𝑇 ) × ℝ𝑑 × ℝ and let 𝜑 be a smooth function such that
max
(𝜒∗ − 𝜑) = (𝜒∗ − 𝜑)(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0.
[0,𝑇 )×ℝ𝑑 ×ℝ
In order to prove that (−∂𝑡 + 𝐻∗ )𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ≤ 0, we assume for contradiction that
(−∂𝑡 + 𝐻∗ )𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) > 0.
(3.13)
An argument analogous to the proof of (3.9) shows that 𝜒∗ (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 1. Consider a
sequence (𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 , 𝜀𝑛 )𝑛≥1 in [0, 𝑇 ) × ℝ𝑑 × ℝ × (0, 1) such that
(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 − 𝜀𝑛 , 𝜀𝑛 ) → (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 , 0)
and 𝜒(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 − 𝜀𝑛 ) → 𝜒∗ (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 1.
Since 𝜒 takes values in {0, 1}, we must have
𝜒(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 − 𝜀𝑛 ) = 1
(3.14)
for all 𝑛 large enough. Set
𝜑(𝑡,
˜ 𝑧, 𝑝) := 𝜑(𝑡, 𝑧, 𝑝) + ∣𝑡 − 𝑡𝑜 ∣2 + ∣𝑧 − 𝑧𝑜 ∣4 + ∣𝑝 − 𝑝𝑜 ∣4 .
Then the inequality (3.13) and the definition of 𝐻∗ imply that we can find (ˆ
𝑢, 𝑎
ˆ) in
𝑁𝐿𝑖𝑝 (⋅, 𝐷𝜑)(𝑡
˜ 𝑜 , 𝑧𝑜 , 𝑝𝑜 ) such that
(
)
(ˆ
𝑢,ˆ
𝑎)(⋅,𝐷 𝜑,𝑣),𝑣
˜
inf −ℒ(𝑍,𝑃 )
𝜑˜ ≥ 0 on 𝐵𝜀 := 𝐵𝜀 (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ,
(3.15)
𝑣∈𝑉
for some 𝜀 > 0. By the definition of 𝑁𝐿𝑖𝑝 , after possibly changing 𝜀 > 0, we have
(ˆ
𝑢, 𝑎
ˆ)(⋅, 𝐷𝜑,
˜ ⋅) ∈ 𝒩0 (⋅, 𝐷𝜑,
˜ ⋅)
on 𝐵𝜀 × 𝑉.
(3.16)
Moreover, we have
𝜑˜ ≥ 𝜑 + 𝜂
on ∂𝐵𝜀
(3.17)
for some 𝜂 > 0. Since 𝜑(𝑡
˜ 𝑛 , 𝑧𝑛 , 𝑝𝑛 ) → 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 𝜒∗ (𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 1, we can find 𝑛
such that
𝜑(𝑡
˜ 𝑛 , 𝑧𝑛 , 𝑝𝑛 ) ≤ 1 + 𝜂/2
(3.18)
19
and such that (3.14) is satisfied. We fix this 𝑛 for the remainder of the proof.
For brevity, we write (ˆ
𝑢, 𝑎
ˆ)(𝑡, 𝑧, 𝑝, 𝑣) for (ˆ
𝑢, 𝑎
ˆ)(𝑧, 𝐷𝜑(𝑡,
˜ 𝑧, 𝑝), 𝑣) in the sequel. Exˆ)[⋅] : 𝒱 → 𝒰 × 𝒜
ploiting the definition of 𝑁𝐿𝑖𝑝 , we can then define the mapping (ˆ
𝔲, 𝔞
implicitly via
(
)
ˆ[𝜈],𝜈
ˆ[𝜈]
𝔲
𝔞
ˆ)[𝜈] = (ˆ
(ˆ
𝔲, 𝔞
𝑢, 𝑎
ˆ) ⋅, 𝑍𝑡𝑛 ,𝑧𝑛 , 𝑃𝑡𝑛 ,𝑝𝑛 , 𝜈 1[𝑡𝑛 ,𝜏 𝜈 ] ,
where
)
}
{
(
ˆ[𝜈],𝜈
ˆ[𝜈]
𝔲
𝔞
𝜏 𝜈 := inf 𝑟 ≥ 𝑡𝑛 : 𝑟, 𝑍𝑡𝑛 ,𝑧𝑛 (𝑟), 𝑃𝑡𝑛 ,𝑝𝑛 (𝑟) ∈
/ 𝐵𝜀 .
ˆ and 𝔞
ˆ are non-anticipating; that is, (ˆ
ˆ) ∈ 𝔘 × 𝔄. Let us write
We observe that 𝔲
𝔲, 𝔞
ˆ[𝜈]
𝔞
ˆ,𝜈
𝔲
𝜈
𝜈
(𝑍 , 𝑃 ) for (𝑍𝑡𝑛 ,𝑧𝑛 , 𝑃𝑡𝑛 ,𝑝𝑛 ) to alleviate the notation. Since 𝜒 ≤ 𝜒∗ ≤ 𝜑, the continuity
of the paths of 𝑍 𝜈 and 𝑃 𝜈 and (3.17) lead to
𝜑 (𝜏 𝜈 , 𝑍 𝜈 (𝜏 𝜈 ), 𝑃 𝜈 (𝜏 𝜈 )) ≤ 𝜑˜ (𝜏 𝜈 , 𝑍 𝜈 (𝜏 𝜈 ), 𝑃 𝜈 (𝜏 𝜈 )) − 𝜂.
On the other hand, in view of (3.15) and (3.16), Itô’s formula applied to 𝜑˜ on [𝑡𝑛 , 𝜏 𝜈 ]
yields that
𝜑˜ (𝜏 𝜈 , 𝑍 𝜈 (𝜏 𝜈 ), 𝑃 𝜈 (𝜏 𝜈 )) ≤ 𝜑˜ (𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 ) .
Therefore, the previous inequality and (3.18) show that
𝜑 (𝜏 𝜈 , 𝑍 𝜈 (𝜏 𝜈 ), 𝑃 𝜈 (𝜏 𝜈 )) ≤ 𝜑˜ (𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 ) − 𝜂 ≤ 1 − 𝜂/2.
By (GDP2𝜒 ) of Lemma 3.3, we deduce that 𝜒(𝑡𝑛 , 𝑧𝑛 , 𝑝𝑛 − 𝜀𝑛 ) = 0, which contradicts (3.14).
To complete the proof of the theorem, we still need to show Lemma 3.5. To this
end, we first make the following observation.
∫
Lemma 3.6. Let 𝛼 ∈ 𝐿2𝑙𝑜𝑐 (𝑊 ) be such that 𝑀 = 𝛼 𝑑𝑊 is a bounded martingale and
let 𝛽 be an ℝ𝑑 -valued, progressively measurable process
such that ∣𝛽∣ ≤ 𝑐(1 + ∣𝛼∣) for
∫
some constant 𝑐. Then the stochastic exponential ℰ( 𝛽 𝑑𝑊 ) is a true martingale.
∫𝑇
Proof. The assumption clearly implies that 0 ∣𝛽𝑠 ∣2 𝑑𝑠 < ∞ ℙ-a.s. Since 𝑀 is bounded,
we have in particular that 𝑀 ∈ 𝐵𝑀 𝑂; i.e.,
[∫
]
𝑇
sup 𝔼
∣𝛼𝑠 ∣2 𝑑𝑠 ∣ℱ𝜏 < ∞.
𝜏 ∈𝒯0
𝜏
∞
∫
In view of the assumption, the
∫ same holds with 𝛼 replaced by 𝛽, so that 𝛽 𝑑𝑊 is in
𝐵𝑀 𝑂. This implies that ℰ( 𝛽 𝑑𝑊 ) is a true martingale; cf. [14, Theorem 2.3].
Proof of Lemma 3.5. Consider the process
𝛽𝑛 (𝑟) :=
𝛿𝑛 (𝑟)
Σ𝑛 (𝑟)1𝐴𝑛 (𝑟);
∣Σ𝑛 (𝑟)∣2
we show that
∣𝛽𝑛 ∣ ≤ 𝑐(1 + ∣𝛼𝑛,𝑣𝑜 ∣)
on [[𝑡𝑛 , 𝜏𝑛 ]]
(3.19)
for some 𝑐 > 0. Then, the result will follow by applying Lemma 3.6 to 𝛼𝑛,𝑣𝑜 1[[𝑡𝑛 ,𝜏𝑛 ]] ;
note that the stochastic integral of this process is bounded by the definition of 𝜏𝑛 . To
prove (3.19), we distinguish two cases.
Case 1: ∂𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ∕= 0. Using that 𝜇 and 𝜎 are continuous and that 𝑈 and 𝐵𝜀 are
bounded, tracing the definitions yields that
{
}
∣𝛿𝑛 ∣ ≤ 𝑐 1 + ∣𝛼𝑛,𝑣𝑜 ∣ + ∣𝛼𝑛,𝑣𝑜 ∣2 ∣∂𝑝𝑝 𝜑(⋅, 𝑍 𝑛 , 𝑃 𝑛 )∣
on [[𝑡𝑛 , 𝜏𝑛 ]],
20
while
∣Σ𝑛 ∣ ≥ −𝑐 + ∣𝛼𝑛,𝑣𝑜 ∣∣∂𝑝 𝜑(⋅, 𝑍 𝑛 , 𝑃 𝑛 )∣ on [[𝑡𝑛 , 𝜏𝑛 ]],
for some 𝑐 > 0. Since ∂𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ∕= 0 by assumption, ∂𝑝 𝜑 is uniformly bounded away
from zero on 𝐵𝜀 , after diminishing 𝜀 > 0 if necessary. Hence, recalling (3.12), there is a
cancelation between ∣𝛿𝑛 ∣ and ∣Σ𝑛 ∣ which allows us to conclude (3.19).
Case 2: ∂𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0. We first observe that
𝛿𝑛+ ≤ 𝑐(1 + ∣𝛼𝑛,𝑣𝑜 ∣) − 𝑐−1 ∣𝛼𝑛,𝑣𝑜 ∣2 ∂𝑝𝑝 𝜑(⋅, 𝑍 𝑛 , 𝑃 𝑛 )
on [[𝑡𝑛 , 𝜏𝑛 ]]
for some 𝑐 > 0. Since 𝛿𝑛− and ∣Σ𝑛 ∣−1 are uniformly bounded on 𝐴𝑛 , it therefore suffices
to show that ∂𝑝𝑝 𝜑 ≥ 0 on 𝐵𝜀 . To see this, we note that (3.6) and the relaxation in the
definition (3.3) of 𝐻 ∗ imply that there exists 𝜄 > 0 such that, for some 𝑣 ∈ 𝑉 and all
small 𝜀 > 0,
− ∂𝑡 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) + 𝐹 (Θ𝜄 , 𝑢, 𝑎, 𝑣) ≤ −𝜂
for all (𝑢, 𝑎) ∈ 𝒩𝜀 (Θ𝜄 ),
(3.20)
where Θ𝜄 = (𝑧0 , 𝑝0 , 𝐷𝜑, 𝐴𝜄 ) and 𝐴𝜄 is the same matrix as 𝐷2 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) except that
the entry ∂𝑝𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) is replaced by ∂𝑝𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) − 𝜄. Going back to the definition
of 𝒩𝜀 , we observe that 𝒩𝜀 (Θ𝜄 ) does not depend on 𝜄 and, which is the crucial part, the
assumption that ∂𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) = 0 implies that 𝒩𝜀 (Θ𝜄 ) is of the form 𝒩 𝑈 × ℝ𝑑 ; that
is, the variable 𝑎 is unconstrained. Now (3.20) and the last observation show that
−(∂𝑝𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) − 𝜄)∣𝑎∣2 ≤ 𝑐(1 + ∣𝑎∣)
for all 𝑎 ∈ ℝ𝑑 , so we deduce that ∂𝑝𝑝 𝜑(𝑡𝑜 , 𝑧𝑜 , 𝑝𝑜 ) ≥ 𝜄 > 0. Thus, after diminishing 𝜀 > 0
if necessary, we have ∂𝑝𝑝 𝜑 ≥ 0 on 𝐵𝜀 as desired. This completes the proof.
3.3
PDE in the monotone case
We now specialize the setup of Section 3.1 to the case where the state process 𝑍 consists
of a pair of processes (𝑋, 𝑌 ) with values in ℝ𝑑−1 × ℝ and the loss function
ℓ : ℝ𝑑−1 × ℝ → ℝ,
(𝑥, 𝑦) 7→ ℓ(𝑥, 𝑦)
is nondecreasing in the scalar variable 𝑦. This setting, which was previously studied
in [3] for the case without adverse control, will allow for a more explicit description of
Λ which is particularly suitable for applications in mathematical finance.
𝔲,𝜈
𝔲,𝜈
𝔲,𝜈
For (𝑡, 𝑥, 𝑦) ∈ [0, 𝑇 ] × ℝ𝑑−1 × ℝ and (𝔲, 𝜈) ∈ 𝒰 × 𝒱, let 𝑍𝑡,𝑥,𝑦
= (𝑋𝑡,𝑥
, 𝑌𝑡,𝑥,𝑦
) be the
strong solution of (3.1) with
(
)
(
)
𝜇𝑋 (𝑥, 𝑢, 𝑣)
𝜎𝑋 (𝑥, 𝑢, 𝑣)
𝜇(𝑥, 𝑦, 𝑢, 𝑣) :=
, 𝜎(𝑥, 𝑦, 𝑢, 𝑣) :=
,
𝜇𝑌 (𝑥, 𝑦, 𝑢, 𝑣)
𝜎𝑌 (𝑥, 𝑦, 𝑢, 𝑣)
where 𝜇𝑌 and 𝜎𝑌 take values in ℝ and ℝ1×𝑑 , respectively. The assumptions from
Section 3.1 remain in force; in particular, the continuity and growth assumptions on 𝜇
and 𝜎. In this setup, we can consider the real-valued function
𝛾(𝑡, 𝑥, 𝑝) := inf{𝑦 ∈ ℝ : (𝑥, 𝑦, 𝑝) ∈ Λ(𝑡)}.
In mathematical finance, this may describe the minimal capital 𝑦 such that the given
𝔲,𝜈
target can be reached by trading in the securities market modeled by 𝑋𝑡,𝑥
; an illustration is given in the subsequent section. In the present context, Corollary 2.3 reads as
follows.
21
Lemma 3.7. Fix (𝑡, 𝑥, 𝑦, 𝑝) ∈ [0, 𝑇 ] × ℝ𝑑−1 × ℝ × ℝ, let 𝑂 ⊂ [0, 𝑇 ] × ℝ𝑑−1 × ℝ × ℝ be
a bounded open set containing (𝑡, 𝑥, 𝑦, 𝑝) and assume that 𝛾 is locally bounded.
(GDP1𝛾 ) Assume that 𝑦 > 𝛾(𝑡, 𝑥, 𝑝 + 𝜀) for some 𝜀 > 0. Then there exist 𝔲 ∈ 𝔘 and
{𝛼𝜈 , 𝜈 ∈ 𝒱} ⊂ 𝒜 such that
(
)
𝔲,𝜈
𝔲,𝜈 𝜈
𝛼𝜈
𝑌𝑡,𝑥,𝑦
(𝜏 𝜈 ) ≥ 𝛾∗ 𝜏, 𝑋𝑡,𝑥
(𝜏 ), 𝑃𝑡,𝑝
(𝜏 𝜈 )
ℙ-a.s. for all 𝜈 ∈ 𝒱,
𝜈
𝔲,𝜈
𝔲,𝜈
𝛼
where 𝜏 𝜈 is the first exit time of (⋅, 𝑋𝑡,𝑥
, 𝑌𝑡,𝑥,𝑦
, 𝑃𝑡,𝑝
) from 𝑂.
(GDP2𝛾 ) Let 𝜑 be a continuous function such that 𝜑 ≥ 𝛾 and let (𝔲, 𝔞) ∈ 𝔘 × 𝔄 and
𝜂 > 0 be such that
(
)
𝔞[𝜈]
𝔲,𝜈
𝔲,𝜈 𝜈
𝑌𝑡,𝑥,𝑦
(𝜏 𝜈 ) ≥ 𝜑 𝜏 𝜈 , 𝑋𝑡,𝑥
(𝜏 ), 𝑃𝑡,𝑝 (𝜏 𝜈 ) + 𝜂 ℙ-a.s. for all 𝜈 ∈ 𝒱,
𝔞[𝜈]
𝔲,𝜈
𝔲,𝜈
where 𝜏 𝜈 is the first exit time of (⋅, 𝑋𝑡,𝑥
, 𝑌𝑡,𝑥,𝑦
, 𝑃𝑡,𝑝 ) from 𝑂. Then 𝑦 ≥ 𝛾(𝑡, 𝑥, 𝑝 − 𝜀)
for all 𝜀 > 0.
Proof. Noting that 𝑦 > 𝛾(𝑡, 𝑥, 𝑝) implies (𝑥, 𝑦, 𝑝) ∈ Λ(𝑡) and that (𝑥, 𝑦, 𝑝) ∈ Λ(𝑡) implies
𝑦 ≥ 𝛾(𝑡, 𝑥, 𝑝), the result follows from Corollary 2.3 by arguments similar to the proof of
Lemma 3.3.
The Hamiltonians 𝐺∗ and 𝐺∗ for the PDE describing 𝛾 are defined like 𝐻 ∗ and 𝐻∗
in (3.3) and (3.4), but with
{
]}
1 [
⊤
⊤
𝐹 (Θ, 𝑢, 𝑎, 𝑣) := 𝜇𝑌 (𝑥, 𝑦, 𝑢, 𝑣) − 𝜇(𝑋,𝑃 ) (𝑥, 𝑢, 𝑣) 𝑞 − Tr 𝜎(𝑋,𝑃 ) 𝜎(𝑋,𝑃 ) (𝑥, 𝑢, 𝑎, 𝑣)𝐴
2
where Θ := (𝑥, 𝑦, 𝑞, 𝐴) ∈ ℝ𝑑−1 × ℝ × ℝ𝑑 × 𝕊𝑑 and
(
)
(
)
𝜇𝑋 (𝑥, 𝑢, 𝑣)
𝜎𝑋 (𝑥, 𝑢, 𝑣)
𝜇(𝑋,𝑃 ) (𝑥, 𝑢, 𝑎, 𝑣) :=
, 𝜎(𝑋,𝑃 ) (𝑥, 𝑢, 𝑎, 𝑣) :=
,
0
𝑎
with the relaxed kernel 𝒩𝜀 replaced by
{
}
𝒦𝜀 (𝑥, 𝑦, 𝑞, 𝑣) := (𝑢, 𝑎) ∈ 𝑈 × ℝ𝑑 : 𝜎𝑌 (𝑥, 𝑦, 𝑢, 𝑣) − 𝑞 ⊤ 𝜎(𝑋,𝑃 ) (𝑥, 𝑢, 𝑎, 𝑣) ≤ 𝜀 ,
and 𝑁𝐿𝑖𝑝 replaced by a set 𝐾𝐿𝑖𝑝 , defined like 𝑁𝐿𝑖𝑝 but in terms of 𝒦0 instead of 𝒩0 .
We then have the following result for the semicontinuous envelopes 𝛾 ∗ and 𝛾∗ of 𝛾.
Theorem 3.8. Assume that 𝛾 is locally bounded. Then 𝛾∗ is a viscosity supersolution
on [0, 𝑇 ) × ℝ𝑑−1 × ℝ of
(−∂𝑡 + 𝐺∗ )𝜑 ≥ 0
and 𝛾 ∗ is a viscosity subsolution on [0, 𝑇 ) × ℝ𝑑−1 × ℝ of
(−∂𝑡 + 𝐺∗ )𝜑 ≤ 0.
Proof. The result follows from Lemma 3.7 by adapting the proof of [3, Theorem 2.1],
using the arguments from the proof of Theorem 3.4 to account for the game-theoretic
setting and the relaxed formulation of the GDP. We therefore omit the details.
We shall not discuss in this generality the boundary conditions as 𝑡 → 𝑇 ; they are
somewhat complicated to state but can be deduced similarly as in [3]. Obtaining a
comparison theorem at the present level of generality seems difficult, mainly due to
the presence of the sets 𝒦𝜀 and 𝐾𝐿𝑖𝑝 (which depend on the solution itself) and the
discontinuity of the nonlinearities at ∂𝑝 𝜑 = 0. It seems more appropriate to treat this
question on a case-by-case basis. In fact, once 𝐺∗ = 𝐺∗ (see also Remark 3.9), the
challenges in proving comparison are similar as in the case without adverse player. For
that case, comparison results have been obtained, e.g., in [5] for a specific setting (see
also the references therein for more examples).
22
Remark 3.9. Let us discuss briefly the question whether 𝐺∗ = 𝐺∗ . We shall focus on
the case where 𝑈 is convex and the (nondecreasing) function 𝛾 is strictly increasing with
respect to 𝑝; in this case, we are interested only in test functions 𝜑 with ∂𝑝 𝜑 > 0. Under
this condition, (𝑢, 𝑎) ∈(𝒦𝜀 (⋅, 𝜑, (∂𝑥 𝜑, ∂𝑝 𝜑), 𝑣) if and only if )there exists 𝜁 with ∣𝜁∣ ≤ 1
such that 𝑎 = (∂𝑝 𝜑)−1 𝜎𝑌 (⋅, 𝜑, 𝑢, 𝑣) − ∂𝑥 𝜑⊤ 𝜎𝑋 (⋅, 𝑢, 𝑣) − 𝜀𝜁 . From this, it is not hard
to see that for such functions, the relaxation 𝜀 ↘ 0, Θ′ → Θ in (3.3) is superfluous as
the operator is already continuous, so we are left with the question whether
inf
sup
𝑣∈𝑉 (𝑢,𝑎)∈𝒦0 (Θ,𝑣)
𝐹 (Θ, 𝑢, 𝑎, 𝑣) =
sup
inf 𝐹 (Θ, 𝑢
ˆ(Θ, 𝑣), 𝑎
ˆ(Θ, 𝑣), 𝑣).
(ˆ
𝑢,ˆ
𝑎)∈𝐾𝐿𝑖𝑝 (Θ) 𝑣∈𝑉
The inequality “≥” is clear. The converse inequality will hold if, say, for each 𝜀 > 0,
there exists a locally Lipschitz mapping (ˆ
𝑢𝜀 , 𝑎
ˆ𝜀 ) ∈ 𝐾𝐿𝑖𝑝 such that
𝐹 (⋅, (ˆ
𝑢𝜀 , 𝑎
ˆ𝜀 )(⋅, 𝑣), 𝑣) ≥
𝐹 (⋅, 𝑢, 𝑎, 𝑣) − 𝜀 for all 𝑣 ∈ 𝑉.
sup
(𝑢,𝑎)∈𝒦0 (⋅,𝑣)
Conditions for the existence of 𝜀-optimal continuous selectors can be found in [15, Theorem 3.2]. If (𝑢𝜀 , 𝑎𝜀 ) is an 𝜀-optimal continuous selector, the definition of 𝒦0 entails
⊤
that 𝑎⊤
𝜀 (Θ, 𝑣)𝑞𝑝 = −𝜎𝑋 (𝑥, 𝑢𝜀 (Θ, 𝑣), 𝑣)𝑞𝑥 + 𝜎𝑌 (𝑥, 𝑦, 𝑢𝜀 (Θ, 𝑣), 𝑣), where we use the notation Θ = (𝑥, 𝑦, 𝑝, (𝑞𝑥⊤ , 𝑞𝑝 )⊤ , 𝐴). Then 𝑢𝜀 can be further approximated, uniformly
on compact sets, by a locally Lipschitz function 𝑢
ˆ𝜀 . We may restrict our attention
⊤
to 𝑞𝑝 > 0; so that, if we
assume
that
𝜎
is
(jointly)
locally Lipschitz,
the map)
( ⊤
−1
ˆ𝜀 (Θ, 𝑣), 𝑣)𝑞𝑥 + 𝜎𝑌 (𝑥, 𝑦, 𝑢
ˆ𝜀 (Θ, 𝑣), 𝑣) is locally Lips−𝜎𝑋 (𝑥, 𝑢
ping 𝑎
ˆ⊤
𝜀 (Θ, 𝑣) := (𝑞𝑝 )
chitz and then (ˆ
𝑢𝜀 , 𝑎
ˆ𝜀 ) defines a sufficiently good, locally Lipschitz continuous selector:
for all 𝑣 ∈ 𝑉 ,
𝐹 (⋅, (ˆ
𝑢𝜀 , 𝑎
ˆ𝜀 )(⋅, 𝑣), 𝑣) ≥ 𝐹 (⋅, (𝑢𝜀 , 𝑎𝜀 )(⋅, 𝑣), 𝑣) − 𝑂𝜀 (1) ≥
sup
𝐹 (⋅, 𝑢, 𝑎, 𝑣) − 𝜀 − 𝑂𝜀 (1)
(𝑢,𝑎)∈𝒦0
on a neighborhood of Θ, where 𝑂𝜀 (1) → 0 as 𝜀 → 0. One can similarly discuss other
cases; e.g, when 𝛾 is strictly concave (instead of increasing) with respect to 𝑝 and
⊤
(𝑥, 𝑢, 𝑣)𝑞𝑥 + 𝜎𝑌 (𝑥, 𝑦, 𝑢, 𝑣) is invertible in 𝑢, with an
the mapping (𝑥, 𝑦, 𝑞𝑥 , 𝑢, 𝑣) 7→ −𝜎𝑋
inverse that is locally Lipschitz, uniformly in 𝑣.
4
Application to hedging under uncertainty
In this section, we illustrate our general results in a concrete example, and use the
opportunity to show how to extend them to a case with unbounded strategies. To this
end, we shall consider a problem of partial hedging under Knightian uncertainty. More
precisely, the uncertainty concerns the drift and volatility coefficients of the risky asset
and we aim at controlling a function of the hedging error; the corresponding worst-case
analysis is equivalent to a game where the adverse player chooses the coefficients. This
problem is related to the 𝐺-expectation of [22, 23], the second order target problem
of [26] and the problem of optimal arbitrage studied in [10]. We let
𝑉 = [𝜇, 𝜇] × [𝜎, 𝜎]
be the possible values of the coefficients, where 𝜇 ≤ 0 ≤ 𝜇 and 𝜎 ≥ 𝜎 ≥ 0. Moreover,
𝑈 = ℝ will be the possible values for the investment policy, so that, in contrast to the
previous sections, 𝑈 is not bounded.
The notation is the same as in the previous section, except for an integrability condition for the strategies that will be introduced below to account for the unboundedness
of 𝑈 , moreover, we shall sometimes write 𝜈 = (𝜇, 𝜎) for an adverse control 𝜈 ∈ 𝒱. Given
𝔲,𝜈
𝔲,𝜈
𝜈
(𝜇, 𝜎) ∈ 𝒱 and 𝔲 ∈ 𝔘, the state process 𝑍𝑡,𝑥,𝑦
= (𝑋𝑡,𝑥
, 𝑌𝑡,𝑦
) is governed by
𝜈
𝑑𝑋𝑡,𝑥
(𝑟)
= 𝜇𝑟 𝑑𝑟 + 𝜎𝑟 𝑑𝑊𝑟 ,
𝜈
𝑋𝑡,𝑥 (𝑟)
23
𝜈
𝑋𝑡,𝑥
(𝑡) = 𝑥
and
(
)
𝔲,𝜈
𝑑𝑌𝑡,𝑦
(𝑟) = 𝔲[𝜈]𝑟 𝜇𝑟 𝑑𝑟 + 𝜎𝑟 𝑑𝑊𝑟 ,
𝔲,𝜈
𝑌𝑡,𝑦
(𝑡) = 𝑦.
𝜈
To wit, the process 𝑋𝑡,𝑥
represents the price of a risky asset with unknown drift and
𝔲,𝜈
volatility coefficients (𝜇, 𝜎), while 𝑌𝑡,𝑦
stands for the wealth process associated to an
investment policy 𝔲[𝜈], denominated in monetary amounts. (The interest rate is zero
for simplicity.) We remark that it is clearly necessary to use strategies in this setup:
even a simple stop-loss investment policy cannot be implemented as a control.
Our loss function is of the form
ℓ(𝑥, 𝑦) = Ψ(𝑦 − 𝑔(𝑥)),
where Ψ, 𝑔 : ℝ → ℝ are continuous functions of polynomial growth. The function Ψ is
also assumed to be strictly increasing and concave, with an inverse Ψ−1 : ℝ → ℝ that
is again of polynomial growth. As a consequence, ℓ is continuous and (2.3) is satisfied
for some 𝑞 > 0; that is,
∣ℓ(𝑧)∣ ≤ 𝐶(1 + ∣𝑧∣𝑞 ),
𝑧 = (𝑥, 𝑦) ∈ ℝ2 .
(4.1)
𝜈
We interpret 𝑔(𝑋𝑡,𝑥
(𝑇 )) as the random payoff of a European option written on the risky
asset, for a given realization of the drift and volatility processes, while Ψ quantifies the
𝔲,𝜈
𝜈
(𝑇 )). In this setup,
disutility of the hedging error 𝑌𝑡,𝑦
(𝑇 ) − 𝑔(𝑋𝑡,𝑥
{
[
]
}
𝔲,𝜈
𝜈
𝛾(𝑡, 𝑥, 𝑝) = inf 𝑦 ∈ ℝ : ∃ 𝔲 ∈ 𝔘 s.t. 𝔼 Ψ(𝑌𝑡,𝑦
(𝑇 ) − 𝑔(𝑋𝑡,𝑥
(𝑇 ))∣ℱ𝑡 ≥ 𝑝 ℙ-a.s. ∀ 𝜈 ∈ 𝒱
is the minimal price for the option allowing to find a hedging policy such that the
expected disutility of the hedging error is controlled by 𝑝.
We fix a finite constant 𝑞¯ > 𝑞 ∨ 2 and define 𝔘 to be the set of mappings 𝔲 : 𝒱 → 𝒰
that are non-anticipating (as in Section 3) and satisfy the integrability condition
⎡
𝑞2¯ ⎤
∫ 𝑇
sup 𝔼 ⎣
∣𝔲[𝜈]𝑟 ∣2 𝑑𝑟 ⎦ < ∞.
(4.2)
0
𝜈∈𝒱
The conclusions below do not depend on the choice of 𝑞¯. The main result of this section
is an explicit expression for the price 𝛾(𝑡, 𝑥, 𝑝).
Theorem 4.1. Let (𝑡, 𝑥, 𝑝) ∈ [0, 𝑇 ] × (0, ∞) × ℝ. Then 𝛾(𝑡, 𝑥, 𝑝) is finite and given by
[ ( 𝜈
)]
𝛾(𝑡, 𝑥, 𝑝) = sup 𝔼 𝑔 𝑋𝑡,𝑥
(𝑇 ) + Ψ−1 (𝑝), where 𝒱 0 = {(𝜇, 𝜎) ∈ 𝒱 : 𝜇 ≡ 0}. (4.3)
𝜈∈𝒱 0
In particular, 𝛾(𝑡, 𝑥, 𝑝) coincides with the superhedging price for the shifted option
𝑔(⋅) + Ψ−1 (𝑝) in the (driftless) uncertain volatility model for [𝜎, 𝜎]; see also below. That
is, the drift uncertainty has no impact on the price, provided that 𝜇 ≤ 0 ≤ 𝜇. Let us
remark, in this respect, that the present setup corresponds to an investor who knows the
present and historical drift and volatility of the underlying. It may also be interesting to
study the case where only the trajectories of the underlying (and therefore the volatility,
but not necessarily the drift) are observed. This, however, does not correspond to the
type of game studied in this paper.
4.1
Proof of Theorem 4.1
Proof of “≥” in (4.3). We may assume that 𝛾(𝑡, 𝑥, 𝑝) < ∞. Let 𝑦 > 𝛾(𝑡, 𝑥, 𝑝); then
there exists 𝔲 ∈ 𝔘 such that
[ ( 𝔲,𝜈
( 𝜈
))]
𝔼 Ψ 𝑌𝑡,𝑦
(𝑇 ) − 𝑔 𝑋𝑡,𝑥
(𝑇 )
≥ 𝑝 for all 𝜈 ∈ 𝒱.
24
As Ψ is concave, it follows by Jensen’s inequality that
( [ 𝔲,𝜈
( 𝜈
)])
Ψ 𝔼 𝑌𝑡,𝑦
(𝑇 ) − 𝑔 𝑋𝑡,𝑥
(𝑇 )
≥ 𝑝 for all 𝜈 ∈ 𝒱.
𝔲,𝜈
Since the integrability condition (4.2) implies that 𝑌𝑡,𝑦
is a martingale for all 𝜈 ∈ 𝒱 0 ,
we conclude that
(
[ ( 𝜈
)])
Ψ 𝑦 − 𝔼 𝑔 𝑋𝑡,𝑥
(𝑇 )
≥ 𝑝 for all 𝜈 ∈ 𝒱 0
[ ( 𝜈
)]
(𝑇 ) + Ψ−1 (𝑝). As 𝑦 > 𝛾(𝑡, 𝑥, 𝑝) was arbitrary, the
and hence 𝑦 ≥ sup𝜈∈𝒱 0 𝔼 𝑔 𝑋𝑡,𝑥
claim follows.
We shall use Theorem 3.8 to derive the missing inequality in (4.3). Since 𝑈 = ℝ is
unbounded, we introduce a sequence of approximating problems 𝛾𝑛 defined like 𝛾, but
with strategies bounded by 𝑛:
{
[ ( 𝔲,𝜈
)
]
}
𝛾𝑛 (𝑡, 𝑥, 𝑝) := inf 𝑦 ∈ ℝ : ∃ 𝔲 ∈ 𝔘𝑛 s.t. 𝔼 ℓ 𝑍𝑡,𝑥,𝑦
(𝑇 ) ∣ℱ𝑡 ≥ 𝑝 ℙ-a.s. ∀ 𝜈 ∈ 𝒱 ,
where
𝔘𝑛 = {𝔲 ∈ 𝔘 : ∣𝔲[𝜈]∣ ≤ 𝑛 for all 𝜈 ∈ 𝒱}.
Then clearly 𝛾𝑛 is decreasing in 𝑛 and
𝛾𝑛 ≥ 𝛾,
𝑛 ≥ 1.
(4.4)
Lemma 4.2. Let (𝑡, 𝑧) ∈ [0, 𝑇 ] × (0, ∞) × ℝ, 𝔲 ∈ 𝔘, and define 𝔲𝑛 ∈ 𝔘 by
𝜈 ∈ 𝒱.
𝔲𝑛 [𝜈] := 𝔲[𝜈]1{∣𝔲[𝜈]∣≤𝑛} ,
Then
[ ( 𝔲𝑛 ,𝜈
)
( 𝔲,𝜈
)
]
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 → 0
ess sup 𝔼 ℓ 𝑍𝑡,𝑧
in 𝐿1 as 𝑛 → ∞.
𝜈∈𝒱
Proof. Using monotone convergence and an argument as in the proof of Step 1 in Section 2.3, we obtain that
}
{
[ ( 𝔲𝑛 ,𝜈
)
( 𝔲,𝜈
)}
{ ( 𝔲𝑛 ,𝜈
)
( 𝔲,𝜈
)
]
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) .
𝔼 ess sup 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) ∣ℱ𝑡 = sup 𝔼 ℓ 𝑍𝑡,𝑧
𝜈∈𝒱
𝜈∈𝒱
Since 𝑉 is bounded, the Burkholder-Davis-Gundy inequalities show that there is a
universal constant 𝑐 > 0 such that
[∫
] 12
𝑇
}
{ 𝔲𝑛 ,𝜈
2
𝔲,𝜈
𝔼 𝑍𝑡,𝑧 (𝑇 ) − 𝑍𝑡,𝑧 (𝑇 )
≤ 𝑐𝔼
∣𝔲[𝜈]𝑟 − 𝔲𝑛 [𝜈]𝑟 ∣ 𝑑𝑟
𝑡
[∫
= 𝑐𝔼
𝑇
] 12
2
𝑑𝑟
𝑟 ∣>𝑛}
𝔲[𝜈]𝑟 1{∣𝔲[𝜈]
𝑡
and hence (4.2) and Hölder’s inequality yield that, for any given 𝛿 > 0,
}
{ 𝔲𝑛 ,𝜈
}
{ 𝔲𝑛 ,𝜈
𝔲,𝜈
𝔲,𝜈
sup ℙ 𝑍𝑡,𝑧
(𝑇 ) − 𝑍𝑡,𝑧
(𝑇 ) > 𝛿 ≤ 𝛿 −1 sup 𝔼 𝑍𝑡,𝑧
(𝑇 ) − 𝑍𝑡,𝑧
(𝑇 ) → 0
𝜈∈𝒱
(4.5)
𝜈∈𝒱
for 𝑛 → ∞. Similarly, the Burkholder-Davis-Gundy inequalities and (4.2) show that
𝔲𝑛 ,𝜈
𝔲,𝜈
{∣𝑍𝑡,𝑧
(𝑇 )∣ + ∣𝑍𝑡,𝑧
(𝑇 )∣, 𝜈 ∈ 𝒱, 𝑛 ≥ 1} is bounded in 𝐿𝑞¯. This yields, on the one hand,
that
𝔲,𝜈
{ 𝔲𝑛 ,𝜈
}
sup ℙ 𝑍𝑡,𝑧
(𝑇 ) + 𝑍𝑡,𝑧
(𝑇 ) > 𝑘 → 0
(4.6)
𝜈∈𝒱, 𝑛≥1
25
for 𝑘 → ∞, and on the other hand, in view of (4.1) and 𝑞¯ > 𝑞, that
{ ( 𝔲𝑛 ,𝜈
)
( 𝔲,𝜈
)
}
ℓ 𝑍𝑡,𝑧 (𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) : 𝜈 ∈ 𝒱, 𝑛 ≥ 1
is uniformly integrable.
(4.7)
Let 𝜀 > 0; then (4.6) and (4.7) show that we can choose 𝑘 > 0 such that
]
[ (
)
( 𝔲,𝜈
)
𝔲𝑛 ,𝜈
𝔲𝑛 ,𝜈
𝔲,𝜈
sup 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) 1{∣𝑍𝑡,𝑧
(𝑇 )∣+∣𝑍𝑡,𝑧 (𝑇 )∣>𝑘} < 𝜀
𝜈∈𝒱
for all 𝑛. Using also that ℓ is uniformly continuous on {∣𝑧∣ ≤ 𝑘}, we thus find 𝛿 > 0
such that
[ ( 𝔲𝑛 ,𝜈
)
( 𝔲,𝜈
)]
sup 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) 𝜈∈𝒱
]
[ (
)
( 𝔲,𝜈
)
𝔲𝑛 ,𝜈
𝔲𝑛 ,𝜈
𝔲,𝜈
≤ 2𝜀 + sup 𝔼 ℓ 𝑍𝑡,𝑧
(𝑇 ) − ℓ 𝑍𝑡,𝑧
(𝑇 ) 1{∣𝑍𝑡,𝑧
(𝑇 )−𝑍𝑡,𝑧
(𝑇 )∣>𝛿} .
𝜈∈𝒱
By (4.5) and (4.7), the supremum on the right-hand side tends to zero as 𝑛 → ∞. This
completes the proof of Lemma 4.2.
Proof of “≤” in (4.3). It follows from the polynomial growth of 𝑔 and the boundedness
of 𝑉 that the right-hand side of (4.3) is finite. Thus, the already established inequality
“≥” in (4.3) yields that 𝛾(𝑡, 𝑥, 𝑝) > −∞. We now show the theorem under the hypothesis that 𝛾(𝑡, 𝑥, 𝑝) < ∞ for all 𝑝; we shall argue at the end of the proof that this is
automatically satisfied.
Step 1: Let 𝛾∞ := inf 𝑛 𝛾𝑛 . Then the upper semicontinuous envelopes of 𝛾 and 𝛾∞
∗
coincide: 𝛾 ∗ = 𝛾∞
.
∗
≥ 𝛾 ∗ . Let 𝜂 > 0 and 𝑦 > 𝛾(𝑡, 𝑥, 𝑝 + 𝜂). We show that
It follows from (4.4) that 𝛾∞
∗
≤ 𝛾 ∗ . Indeed,
𝑦 ≥ 𝛾𝑛 (𝑡, 𝑥, 𝑝) for 𝑛 large; this will imply the remaining inequality 𝛾∞
the definition of 𝛾 and Lemma 4.2 imply that we can find 𝔲 ∈ 𝔘 and 𝔲𝑛 ∈ 𝔘𝑛 such that
𝐽(𝑡, 𝑥, 𝑦, 𝔲𝑛 ) ≥ 𝐽(𝑡, 𝑥, 𝑦, 𝔲) − 𝜖𝑛 ≥ 𝑝 + 𝜂 − 𝜖𝑛
ℙ-a.s.,
where 𝜖𝑛 → 0 in 𝐿1 . If 𝐾𝑛 is defined like 𝐾, but with 𝔘𝑛 instead of 𝔘, then it follows that
𝐾𝑛 (𝑡, 𝑥, 𝑦) ≥ 𝑝 + 𝜂 − 𝜖𝑛 ℙ-a.s. Recalling that 𝐾𝑛 is deterministic (cf. Proposition 3.1),
we may replace 𝜖𝑛 by 𝔼[𝜖𝑛 ] in this inequality. Sending 𝑛 → ∞, we then see that
lim𝑛→∞ 𝐾𝑛 (𝑡, 𝑥, 𝑦) ≥ 𝑝 + 𝜂, and therefore 𝐾𝑛 (𝑡, 𝑥, 𝑦) ≥ 𝑝 + 𝜂/2 for 𝑛 large enough.
The fact that 𝑦 ≥ 𝛾𝑛 (𝑡, 𝑥, 𝑝) for 𝑛 large then follows from the same considerations as in
Lemma 2.4.
Step 2: The relaxed semi-limit
∗
𝛾¯∞
(𝑡, 𝑥, 𝑝) :=
lim sup
𝑛→∞
(𝑡′ ,𝑥′ ,𝑝′ )→(𝑡,𝑥,𝑝)
𝛾𝑛∗ (𝑡′ , 𝑥′ , 𝑝′ )
is a viscosity subsolution on [0, 𝑇 ) × (0, ∞) × ℝ of
{
}
1 2 2
− ∂𝑡 𝜑 + inf
− 𝜎 𝑥 ∂𝑥𝑥 𝜑 ≤ 0
2
𝜎∈[𝜎,𝜎]
(4.8)
∗
and satisfies the boundary condition 𝛾¯∞
(𝑇, 𝑥, 𝑝) ≤ 𝑔(𝑥) + Ψ−1 (𝑝).
We first show that the boundary condition is satisfied. Fix (𝑥, 𝑝) ∈ (0, ∞) × ℝ
and let 𝑦 > 𝑔(𝑥) + Ψ−1 (𝑝); then ℓ(𝑥, 𝑦) > 𝑝. Let (𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) → (𝑇, 𝑥, 𝑝) be such that
∗
𝛾𝑛 (𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) → 𝛾¯∞
(𝑇, 𝑥, 𝑝). We consider the strategy 𝔲 ≡ 0 and use the arguments
from the proof of Proposition 3.1 to find a constant 𝑐 independent of 𝑛 such that
[
]
(
)
𝑞
¯
𝑞¯
¯
2 + ∣𝑥 − 𝑥 ∣𝑞
ess sup 𝔼 ∣𝑍𝑡0,𝜈
(𝑇
)
−
(𝑥,
𝑦)∣
∣ℱ
≤
𝑐
∣𝑇
−
𝑡
∣
.
𝑡
𝑛
𝑛
𝑛
𝑛 ,𝑥𝑛 ,𝑦
𝜈∈𝒱
26
Similarly as in the proof of Lemma 4.2, this implies that there exist constants 𝜀𝑛 → 0
such that
𝐽(𝑡𝑛 , 𝑥𝑛 , 𝑦, 0) ≥ ℓ(𝑥, 𝑦) − 𝜀𝑛 ℙ-a.s.
In view of ℓ(𝑥, 𝑦) > 𝑝, this shows that 𝑦 ≥ 𝛾𝑛 (𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) for 𝑛 large enough, and hence
∗
∗
that 𝑦 ≥ 𝛾¯∞
(𝑇, 𝑥, 𝑝). As a result, we have 𝛾¯∞
(𝑇, 𝑥, 𝑝) ≤ 𝑔(𝑥) + Ψ−1 (𝑝).
It remains to show the subsolution property. Let 𝜑 be a smooth function and let
(𝑡𝑜 , 𝑥𝑜 , 𝑝𝑜 ) ∈ [0, 𝑇 ) × (0, ∞) × ℝ be such that
∗
∗
(¯
𝛾∞
− 𝜑)(𝑡𝑜 , 𝑥𝑜 , 𝑝𝑜 ) = max(¯
𝛾∞
− 𝜑) = 0.
After passing to a subsequence, [1, Lemma 4.2] yields (𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) → (𝑡𝑜 , 𝑥𝑜 , 𝑝𝑜 ) such
that
∗
lim (𝛾𝑛∗ − 𝜑)(𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) = (¯
𝛾∞
− 𝜑)(𝑡𝑜 , 𝑥𝑜 , 𝑝𝑜 ),
𝑛→∞
and such that (𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) is a local maximizer of (𝛾𝑛∗ − 𝜑). Applying Theorem 3.8 to
𝛾𝑛∗ , we deduce that
sup
inf
𝑛 (⋅,𝐷𝜑) (𝜇,𝜎)∈𝑉
(ˆ
𝑢,ˆ
𝑎)∈𝐾𝐿𝑖𝑝
𝐺𝜑(⋅, (ˆ
𝑢, 𝑎
ˆ)(𝜇, 𝜎), (𝜇, 𝜎))(𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) ≤ 0,
(4.9)
where
𝐺𝜑(⋅, (𝑢, 𝑎), (𝜇, 𝜎)) := 𝑢𝜇 − ∂𝑡 𝜑 − 𝜇𝑥∂𝑥 𝜑 −
)
1( 2 2
𝜎 𝑥 ∂𝑥𝑥 𝜑 + 𝑎2 ∂𝑝𝑝 𝜑 + 2𝜎𝑥𝑎∂𝑥𝑝 𝜑
2
𝑛
and 𝐾𝐿𝑖𝑝
(⋅, 𝐷𝜑)(𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) is the set of locally Lipschitz mappings (ˆ
𝑢, 𝑎
ˆ) with values in
[−𝑛, 𝑛] × ℝ such that
𝜎ˆ
𝑢(𝑥, 𝑞𝑥 , 𝑞𝑝 , 𝜇, 𝜎) = 𝑥𝜎𝑞𝑥 + 𝑞𝑝 𝑎
ˆ(𝑥, 𝑞𝑥 , 𝑞𝑝 , 𝜇, 𝜎)
for all 𝜎 ∈ [𝜎, 𝜎]
for all (𝑥, (𝑞𝑥 , 𝑞𝑝 )) in a neighborhood of (𝑥𝑛 , 𝐷𝜑(𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 )). Since the mapping
(0, ∞) × ℝ2 × [𝜇, 𝜇] × [𝜎, 𝜎] → ℝ2 ,
(𝑥, 𝑞𝑥 , 𝑞𝑝 , 𝜇, 𝜎) 7→ (𝑥𝑞𝑥 , 0)
𝑛
belongs to 𝐾𝐿𝑖𝑝
(⋅, 𝐷𝜑)(𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) for 𝑛 large enough, (4.9) leads to
{
−∂𝑡 𝜑 + inf
𝜎∈[𝜎,𝜎]
}
1
− 𝜎 2 𝑥2 ∂𝑥𝑥 𝜑 (𝑡𝑛 , 𝑥𝑛 , 𝑝𝑛 ) ≤ 0
2
for 𝑛 large. Here the nonlinearity is continuous; therefore, sending 𝑛 → ∞ yields (4.8).
∗
Step 3: We have 𝛾¯∞
≤ 𝜋 on [0, 𝑇 ] × (0, ∞) × ℝ, where
[ ( 𝜈
)]
𝜋(𝑡, 𝑥, 𝑝) := sup 𝔼 𝑔 𝑋𝑡,𝑥
(𝑇 ) + Ψ−1 (𝑝)
𝜈∈𝒱 0
is the right hand side of (4.3).
Indeed, our assumptions on 𝑔 and Ψ−1 imply that 𝜋 is continuous with polynomial
growth. It then follows by standard arguments that 𝜋 is a viscosity supersolution on
[0, 𝑇 ) × (0, ∞) × ℝ of
{
}
1 2 2
−∂𝑡 𝜑 + inf
− 𝜎 𝑥 ∂𝑥𝑥 𝜑 ≥ 0,
2
𝜎∈[𝜎,𝜎]
and clearly the boundary condition 𝜋(𝑇, 𝑥, 𝑝) ≥ 𝑔(𝑥) + Ψ−1 (𝑝) is satisfied. The claim
then follows from Step 2 by comparison.
27
We can now deduce the theorem: We have 𝛾 ≤ 𝛾 ∗ by the definition of 𝛾 ∗ and
∗
∗
∗
𝛾 = 𝛾∞
by Step 1. As 𝛾∞
≤ 𝛾¯∞
by construction, Step 3 yields the result.
It remains to show that 𝛾 < ∞. Indeed, this is clearly satisfied when 𝑔 is bounded
from above. For the general case, we consider 𝑔𝑚 = 𝑔 ∧ 𝑚 and let 𝛾𝑚 be the corresponding value function. Given 𝜂 > 0, we have 𝛾𝑚 (𝑡, 𝑥, 𝑝 + 𝜂) < ∞ for all 𝑚 and so (4.3)
holds for 𝑔𝑚 . We see from (4.3) that 𝑦 := 1 + sup𝑚 𝛾𝑚 (𝑡, 𝑥, 𝑝 + 𝜂) is finite. Thus, there
exist 𝔲𝑚 ∈ 𝔘 such that
[ ( 𝔲𝑚 ,𝜈
( 𝜈
))
]
𝔼 Ψ 𝑌𝑡,𝑦
(𝑇 ) − 𝑔𝑚 𝑋𝑡,𝑥
(𝑇 ) ∣ℱ𝑡 ≥ 𝑝 + 𝜂 for all 𝜈 ∈ 𝒱.
∗
Using once more the boundedness of 𝑉 , we see that for 𝑚 large enough,
[ ( 𝔲𝑚 ,𝜈
( 𝜈
))
]
𝔼 Ψ 𝑌𝑡,𝑦
(𝑇 ) − 𝑔 𝑋𝑡,𝑥
(𝑇 ) ∣ℱ𝑡 ≥ 𝑝 for all 𝜈 ∈ 𝒱,
which shows that 𝛾(𝑡, 𝑥, 𝑝) ≤ 𝑦 < ∞.
Remark 4.3. We sketch a probabilistic proof for the inequality “≤” in Theorem 4.1, for
the special case without drift (𝜇 = 𝜇 = 0) and 𝜎 > 0. We focus on 𝑡 = 0 and recall that
𝜈
𝑦0 := sup𝜈∈𝒱 0 𝔼[𝑔(𝑋0,𝑥
(𝑇 ))] is the superhedging price for 𝑔(⋅) in the uncertain volatility
model. More precisely, if 𝐵 is the coordinate-mapping process on Ω = 𝐶([0, 𝑇 ]; ℝ),
there exists an 𝔽𝐵 -progressively measurable process 𝜗 such that
𝑇
∫
𝑦0 +
𝜗𝑠
0
𝑑𝐵𝑠
≥ 𝑔(𝐵𝑇 ) 𝑃 𝜈 -a.s. for all 𝜈 ∈ 𝒱 0 ,
𝐵𝑠
𝜈
where 𝑃 𝜈 is the law of 𝑋0,𝑥
under 𝑃 (see, e.g., [20]). Seeing 𝜗 as an adapted functional
of 𝐵, this implies that
∫
𝑦0 +
𝑇
𝜈
𝜗𝑠 (𝑋0,𝑥
)
0
𝜈
𝑑𝑋0,𝑥
(𝑠)
𝜈
≥ 𝑔(𝑋0,𝑥
(𝑇 ))
𝜈
𝑋0,𝑥 (𝑠)
𝑃 -a.s. for all 𝜈 ∈ 𝒱 0 .
𝜈
𝜈
is non-anticipating with respect to 𝜈, we see that 𝔲[𝜈]𝑠 := 𝜗𝑠 (𝑋0,𝑥
) defines a
Since 𝑋0,𝑥
−1
non-anticipating strategy such that, with 𝑦 := 𝑦0 + Ψ (𝑝),
∫
𝑦+
𝑇
𝔲[𝜈]𝑠
0
𝜈
𝑑𝑋0,𝑥
(𝑠)
𝜈
≥ 𝑔(𝑋0,𝑥
(𝑇 )) + Ψ−1 (𝑝);
𝜈
𝑋0,𝑥 (𝑠)
that is,
( 𝔲,𝜈
)
𝜈
Ψ 𝑌0,𝑦
(𝑇 ) − 𝑔(𝑋0,𝑥
(𝑇 ) ≥ 𝑝
holds even 𝑃 -almost surely, rather than only in expectation, for all 𝜈 ∈ 𝒱 0 , and 𝒱 0 = 𝒱
because of our assumption that 𝜇 = 𝜇 = 0. In particular, we have the existence of an
optimal strategy 𝔲. (We notice that, in this respect, it is important that our definition
of strategies does not contain regularity assumptions on 𝜈 7→ 𝔲[𝜈].)
Heuristically, the case with drift uncertainty (i.e., 𝜇 ∕= 𝜇) can be reduced to the
above by a Girsanov change of measure argument; e.g.,∫if 𝜇 is deterministic, then we
can take 𝔲[(𝜇, 𝜎)] := 𝔲[(0, 𝜎 𝜇 )], where 𝜎 𝜇 (𝜔) := 𝜎(𝜔 + 𝜇𝑡 𝑑𝑡). However, for general
𝜇, there are difficulties related to the fact that a Girsanov Brownian motion need not
generate the original filtration (see, e.g., [9]), and we shall not enlarge on this.
References
[1] G. Barles. Solutions de viscosité des équations de Hamilton-Jacobi. Springer, Paris,
1994.
28
[2] E. Bayraktar and S. Yao. On zero-sum stochastic differential games. Preprint
arXiv:1112.5744v3, 2011.
[3] B. Bouchard, R. Elie, and N. Touzi. Stochastic target problems with controlled
loss. SIAM J. Control Optim., 48(5):3123–3150, 2009.
[4] B. Bouchard and M. Nutz. Weak dynamic programming for generalized state
constraints. SIAM J. Control Optim., 50(6):3344–3373, 2012.
[5] B. Bouchard and T. N. Vu. A stochastic target approach for P&L matching problems. Math. Oper. Res., 37(3):526–558, 2012.
[6] R. Buckdahn, Y. Hu, and J. Li. Stochastic representation for solutions of Isaacs’
type integral-partial differential equations. Stochastic Process. Appl., 121(12):2715–
2750, 2011.
[7] R. Buckdahn and J. Li. Stochastic differential games and viscosity solutions of
Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim., 47(1):444–
475, 2008.
[8] C. Dellacherie and P. A. Meyer. Probabilities and Potential B. North Holland,
Amsterdam, 1982.
[9] J. Feldman and M. Smorodinsky. Simple examples of non-generating Girsanov
processes. In Séminaire de Probabilités XXXI, volume 1655 of Lecture Notes in
Math., pages 247–251. Springer, Berlin, 1997.
[10] D. Fernholz and I. Karatzas. Optimal arbitrage under model uncertainty. Ann.
Appl. Probab., 21(6):2191–2225, 2011.
[11] W. H. Fleming and P. E. Souganidis. On the existence of value functions of twoplayer, zero-sum stochastic differential games. Indiana Univ. Math. J., 38(2):293–
314, 1989.
[12] H. Föllmer and P. Leukert. Quantile hedging. Finance Stoch., 3(3):251–273, 1999.
[13] N. El Karoui and M.-C. Quenez. Dynamic programming and pricing of contingent
claims in an incomplete market. SIAM J. Control Optim., 33(1):29–66, 1995.
[14] N. Kazamaki. Continuous exponential martingales and BMO, volume 1579 of Lecture Notes in Math. Springer, Berlin, 1994.
[15] A. Kucia and A. Nowak. On 𝜖-optimal continuous selectors and their application in
discounted dynamic programming. J. Optim. Theory Appl., 54(2):289–302, 1987.
[16] J. Li and S. Peng. Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman
equations. Nonlinear Anal., 70(4):1776–1796, 2009.
[17] J.-F. Mertens. Théorie des processus stochastiques généraux applications aux surmartingales. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22:45–68, 1972.
[18] L. Moreau. Stochastic target problems with controlled loss in jump diffusion models. SIAM J. Control Optim., 49(6):2577–2607, 2011.
[19] J. Neveu. Discrete-Parameter Martingales. North-Holland, Amsterdam, 1975.
[20] M. Nutz and H. M. Soner. Superhedging and dynamic risk measures under volatility
uncertainty. SIAM J. Control Optim., 50(4):2065–2089, 2012.
29
[21] S. Peng. BSDE and stochastic optimizations. In J. Yan, S. Peng, S. Fang, and
L. Wu, editors, Topics in Stochastic Analysis, chapter 2, Science Press, Beijing (in
Chinese), 1997.
[22] S. Peng. 𝐺-expectation, 𝐺-Brownian motion and related stochastic calculus of
Itô type. In Stochastic Analysis and Applications, volume 2 of Abel Symp., pages
541–567, Springer, Berlin, 2007.
[23] S. Peng. Multi-dimensional 𝐺-Brownian motion and related stochastic calculus
under 𝐺-expectation. Stochastic Process. Appl., 118(12):2223–2253, 2008.
[24] H. M. Soner and N. Touzi. Dynamic programming for stochastic target problems
and geometric flows. J. Eur. Math. Soc. (JEMS), 4(3):201–236, 2002.
[25] H. M. Soner and N. Touzi. Stochastic target problems, dynamic programming, and
viscosity solutions. SIAM J. Control Optim., 41(2):404–424, 2002.
[26] H. M. Soner, N. Touzi, and J. Zhang. Dual formulation of second order target
problems. Ann. Appl. Probab., 23(1):308–347, 2013.
30